Hi all! I wanted to quickly address a misconception that I've been seeing in a lot of comments. A number of you seem to be confused as to why a path is defined using a function, since that would seemingly exclude paths such as circles which fail the vertical line test. The thing to realize is a "path" here is not defined simply as a function that takes a real number input and then outputs another real number, but it outputs points in the space, which for a 2D space means entire (x,y) pairs. So a path in 2D space is a function that takes real number inputs in the range [0,1] and outputs (x,y) pairs, not just y-values. So for example, a full circular path can be defined as f(t) = (cos t, sin t) which describes a point moving counter-clockwise around a circle of radius 1 (if t ranges from 0 to 2pi). If you think of t as representing time, then this literally describes a point moving around a circle with speed 1. I hope that helps!
@@viliml2763 actually, its not always possible to find a parameterization of multivariable functions. and indeed, when you parameterize something you define a "function" not an "equation". so what you said doesnt abide by the actual definitions of parameterizations, at least the ones i have seen.
Ok so the topologists sine curve doesn't have a limit as it approaches zero, or I guess it is undefined. Which makes sense that therefore the curve is "disconnected" by the first definition. But what if we add that line segment stretching from (0,-1) to (0,1)? Does it then become connected? Let's say you picked the points on the curve (-1,sin-1) and (1,sin1) and let them be t = 0 and 1 respectively. As such when t = 0.5 then f should = (0,0) and there is some range of values of t that will return points on the curve with an x value of zero. Right? What have I missed?
Hee! I did my analysis course after studying topology. The assignment grader hated be for solving problems in terms of (eg) 'coverings of open sets'. Three cheers for soft analysis!
@@micayahritchie7158 I"m a programmer and I do a lot of code refactoring -- rewriting expressions so that they are more elegant. I was doing this with the epsilon_felta d definition one day. To my surprise, it brought me to the thteshold of the more general topological definition. The intuition guiding my steps was not mathematical at all, but (as far as I could tell) solely with making the case expression more elegant. It amazed (and amazes) be to think: can concern for (and practice with) code refactoring, over the space of a few hours, replicate a fifty(?) plus year jump From Wierstrass to topology? Might he (who is attributed with the epsilon-felts definition), simply by obsessively 'polishing his code', have been led to considerations of general topology? The thought is tantalizing. What might it entail for research today? Cheers! :)
The structure of this video is fantastic! Every time I thought something was confusing or needed an example, it was immediately followed by further explanation or an example. Anticipating how people are going to follow along is difficult, so I'm very impressed with how well this video manages to do that
Its a good visual for what happens to light near a black hole. Any frequency of light gets redshifted to a wavelength impossible to detect even against the CMB
But I guess it's practically impossible. Like here in this beautiful video it takes roughly 22 minutes to explain the motivation behind just 2 definitions. That's cool but we've encountered hundreds, if not thousands of definitions/theorems in various branches of maths so far from school to master degree level. So I think it's obviously why our education system is the way it is. We don't have infinite time to explain every possible concept the way this guy or 3b1b explains!!!! It just isnt practical
@@randomdude9135 As a biologist I also learned a lot of concepts, but two decades after university I realize that at least half of them could be omitted at the benefit of delving into more depth regarding the more basic ideas. Even at the expense of being less specialized... I would guess that in math to be able to connect between different branches is even more important either for practical or for theoretical creativity. Maybe some specializations teaching and exploring the historical development of individual sciences strictly focusing on the cognitive processes necessary for a fuller understanding of the concepts
The topologist's sine curve is a very cool topological space, and it is a very natural place to get into the nitty-gritty of what connectedness means in all its forms. I think you have done that very well here. However, one point of criticism: I think it's wrong to entirely fail to mention around 5:30 the possibility that there may be *more* than one value t that gives f(t) = (0, y). It doesn't change the result, but it does slightly complicate the argument (you need to choose your t* carefully), so I can understand glossing over it. But not mentioning it at all is far beyond just glossing over it, and I personally think that falls into the territory of "incorrect". You have a note for the pedantic when talking about neighbourhoods at 8:25, that could've been a good fit here too.
I've certainly been known to err, but from what I can understand, I'm not sure I'd glossed over all that much? In the case of multiple possible t* values, I think the basic argument I used works if we take t* to be the minimum such t* (and I think you can justify via continuity of f that such a minimum exists). Then the argument proceeds as normal (though you might have to swap out the two-sided limits for left-handed limits). Is that the subtlety you were referring to?
@@morphocular Yeah, more or less. Continuity says the set of ts where f(t) has zero x-coordinate must be closed, the intermediate value theorem says it must be non-empty, and Heine-Borel (or something) tells you it is compact and therefore contains its minimum. If you pick t* to just be an arbitrary such t, rather than the minimum, then the limiting argument tells you nothing.
@@MasterHigure Gotcha. And thanks for the feedback! Figuring out where to draw the line at how much detail to skip is a never-ending battle for me, so I appreciate your input. To be honest, I hadn't considered that multiple t* values could be much of an issue until I realized (just today!) that there could be a whole interval of them!
@@morphocular There is no objective place to draw the line. Heck, I don't even know that it make sense to speak about *consistency* with where you draw the line, much less correctness. I have told you one place where I disagree with your line drawing, and one where I agree, but that doesn't really mean I'm right and you're wrong.
I just love the ingenuity in that definition! A set is compact if every open cover has a finite subcover. That is actually such a cool definition of a concept that is intuitively so easy.
I love how in the last moments of the video he showed a graphic of the epsilon delta definition of continuity that instantly made me understand it despite eluding me for years at this point. Excellent video if anyone's reading this.
this was such a beautiful video i HAD to leave a comment, like, and subscribe. what i've found really beautiful in mathematics, is when we start coming up with definitions to seemingly answer an extreme case of an intuitive problem, the solution is just "whatever you so desire". it took me sometime to appreciate this beauty, because at first it feels like a slap in the face, like when you're in an adventure story to find the truth of the world, meets an all-knowing-being find what is the meaning of life and they ask back at you "what is your meaning of life?". and this video really encapsulates that feeling. also that 1st textbook definition of "connected" really blew me away. it's almost like they used the aforementioned (two open disks touching at their boundary, a.k.a. a union of two non-empty disjoint open sets) to come up with a definition. fascinating.
Yes. Much of mathematics is about the language to represent concepts. There are definitions, and not knowing them makes it impossible to solve a problem that requires knowing it. But does not make you any more or less smart or intuitive. However, once one finds out and understands a definition, there are many important and useful things that result (e.g., theorems that can be applied to large classes of problems), and then “correct” and “incorrect” answers arise from the definitions (so that our language remains consistent). This last point is sometimes lost on students, who get angry and argue against definitions (they are arbitrary, but must be accepted, like we accept the meaning of words in spoken language… but with much more rigor in mathematics!). This is futile, “by definition,” so don’t do it, and instead focus on understanding the deep meaning of the definitions and their applications. Great video.
Even more so... I think that a sufficiently intelligent being less occupied with limitations of its individual existence would ask back: What do you mean by "meaning" and why is it important for you?
Interesting video. And I can certainly appreciate the point you made at the end about the importance of definitions and the difficulties mathematicians sometimes have formulating good definitions. I recall from my studies of calculus that it took a while for mathematicians to derive the definitions of limit and derivative that we take for granted today. Before our current definition of limit was proposed nobody could define limits in terms that did not verge on the mystic.
eyeopening and excellently narrated. The visuals and transitions were just perfect as well. Perfect video on the subject, thank you for the hard work it took to make this!
Absolutely fantastic video. Even though I encountered this example 5 years ago, I can honestly say this is the first time I actually truly capture the subtle difference between the two definitions. Also, kudos on the definition-motivation plea at the end. Great work!
After watching the intro, here's my guess: to connect the two "ends" of the curve around 0, those individual ends must have well-defined limits. Even though the overall curve does not have a well-defined limit at 0, you could imagine each individual end having one. However, for this curve, each side of the curve near 0 is just as badly-behaved as 0 itself.
For #1, see that if you view the set (say, the two (almost) touching open/closed disks) not as a subset of a larger space but as a space itself, it has no boundary (there are no points outside of it as it is everything), and furthermore every connected component has no boundary and so it is an open set. If it has multiple components then it is the union of disjoint open sets. For #2, see that being both open and closed at the same time means having no boundary (contain all boundary while containing none of the boundary = boundary is nothing)
Great video! I especially like the bit at the end when you compare semantics. I think math's greatest problem is that definitions usually throw away all context and motivation which makes them happen in the first place. It's like trying to describe an algorithm in assembly/binary, yeah it's useful for execution, but beyond terrible for understanding the ideas at play.
I love point-set topology for how completely insane and seemingly contradictory it is despite being entirely consistent. You didn't even get into locally connected and how it's separated from connected. Also, I don't think the textbook definition of connected is that hard to explain, since a closed set is defined to be the complement of an open set. So if a non-trivial set in the space is both open and closed, then it's a non-empty open set, whose complement is also non-empty and open.
15:57 My mind went immediately to the Touching Subsets definition of connectedness and concluded the Topologists' Sine Curve is connected, before you reminded me about the path-connected definition.
My attempt at coming up with a definition for "connectedness," as you mention at 2:30. I wanted to see how I'd do before I see the answer. Define a Connector set as being constructed by the following process. Start with a well ordered set of m points with m>1. Call this the seed set A = {a_i| i from 1 to m inclusive}. The connector set is the union of all of the line segments connecting a_i to a_i+1 for i ranging from 1 to m-1. Now suppose we want to assess if a region R is connected. The set R is connected iff for all r_1 and r_2 in R, there exists a connector set A containing r_1 and r_2 st A is contained within R. Edit: Ah I now see this definition only works if R is of at least 2 dimensions everywhere. Tricky!
I would argue that the curve with the line along the y-axis is still connected with the path definition (Disclaimer: I am not a mathematician, so feel free to correct me if I'm misusing terminology; more notes at bottom). There are only two possible scenarios for the "endpoints" of the separated curves without the vertical line (these "endpoints" reflect the y-value of the function immediately to the left and immediately to the right of x=0). The endpoints are either located at effectively the same point to create what would normally be a removable discontinuity or located at different y-values to create what would normally be a jump discontinuity. These types of discontinuity usually refer to functions where the limits at the x-value are known. In this case, the limits from the left and right are undefined due to the oscillating pattern. However, the function is bounded between two y-values, so we know that the "endpoints" of both sides will each have a y-value between -1 and 1. This would mean that adding a line at x=0 that ranges from y=-1 to y=1 should attach to both disconnected "endpoints" regardless of where they are positioned relative to the y-axis. Although we cannot determine the precise values of the path function as it meets the y-axis, similar to the way that we cannot determine a finite limit at x=0, an indefinite path function must exist that traces the vertical line to connect the two segments. Note: This is the result of a bunch of concepts and theorems back from Calculus 1 that have been scrambled together from memory to try to make something that resembles a decent explanation. The main inspiring concepts aside from discontinuity were the Intermediate Value Theorem and the Squeeze Theorem. It's also 2:30am as I'm finishing this (I got hyperfixated on number stuff again), so please let me know if there's something that doesn't make sense, and I'll try my best to clarify later
Topologist's sine curve is one of my favourite so I watch this! If I don't misunderstand, the definition of the curve T you used is T:=the union of left part L, right part R and Y={(0,y): -1
Very interesting video. But please kindly clarify regarding the Ordered Square (17:41) 1. Why a space-filling curve, such as the (infinite iteration of the) Hilbert curve cannot be considered an ordered square? 2. If it is indeed possible to use the Hilbert curve as a 1D path through the ordered square, then would this also enable us to use the path idea to define connectedness? Thanks in advance to anyone explaining these questions.
I'm not sure, but I think a space filling curve _could_ be used, but in topology, there's always a definition hiding somewhere of what topological space you're dealing with, and that includes a so-called topology. The ordered square is using the order topology for the given ordering, while using a space filling curve would give you a different topology (if even possible, idk), so you'd be dealing with a different topological space, just like how the ordered square is different from the regular square topologically, even though they are the same when only looking at what points there are
It has to be continuous in the order topology, not the usual Euclidean topology. The order topology is the topology where open intervals or any union of open intervals are considered open sets.
Before I watch the rest of the video, you asked for my thoughts. I think the graph... is both connected and disconnected. On one hand, in the same way that there are infinitely many values between 0 and 1, now we find that no matter how far we zoom in on the y-axis, we can always zoom in further and insert a new point on either side that will have a different y-value. There is no meaningful "final" value of y for the "final" position x immediately below and above 0, so there is no well defined "attachment point" to which the y-axis can be "connected" with the curves on either side, hence disconnected. But, this oscillation continues infinitely, so there is no break or discontinuity between the y-axis and the curves, hence connected. Can limits help with this, maybe? As they say on "Hollywoo Stars and Celebrities: What do they know, do they know stuff? Let's find out," let's find out!
I have been studying civil engineering since last October and am struggling to grasp the concept of continuous functions. Thanks to your videos, some things have started to click and the gears are starting to move. That may be as well because I have been studying for the upcoming math exams for the entirety of last week, but hey, i’ll take what I can get.
21:18 “But in any case, one of the things I hope you take away from this video is an appreciation of the ingenuity that goes into crafting a good mathematical definition. So often in Math, the focus is on the big theorems, proofs, and calculations, but I feel we don’t often pause to appreciate all the creativity and cleverness that go into some of the definitions. Maybe it’s partly because textbooks and lectures often just state definitions unmotivated before quickly moving on to the theorems and proofs. *But just as every theorem needs a proof, I think every definition needs a motivation.* And it’s an important mathematical skill to cultivate too! One that I worry too often goed unexercised. So the next time you encounter a strange mathematical definition out in the wild, take a minute to flex that mathematical muscle and try to understand what motivated it, and also appreciate the work that went into crafting it.”
"Just as every theorem needs a proof, every definition needs a motivation" It feels like I'm the only one in my math class thinking this, including the teacher ;)
I'd say they don't agree because the topological definition doesn't include the requirement of a limit, only that the sets share a boundary which is in the whole set, which I interpret to mean the function has (at least one) y value for each x value. Even in the 'path definition', if you remove the requirement that the limit exists, you can define a (discontinuous) function that works (but again that is not the definition of "path")
Another question: instead of the topologist's sine curve, if we modify it such that we only add the point (0, 0) instead of a whole strip from (0, -1) to (0, 1), would the resulting set still be connected?
It still has the same properties as the version in the video (which is the closed topologist sine curve) It's disconnected in some definition but connected in others
It does not even have to be (0, 0). You can add any non-empty subset of the line segment to make it connected, as you just need to guarantee one point of the common boundary in either parts of a split.
Really nice introduction to these concepts! However, an important thing I always try to teach my students is that topological notions are always relative. An Intervall in R may be open, however the same interrval in R^2 will not be w.r.t. the standard topology on R^2. Usually the topologist's sine curve is not just considered as a subset of R^2 with its standard topology, but rather as a topological space in it's own right, by giving it the subspace topology. In this case your argument, that all points of the curve are boundary points, is incorrect. That's also why connectedness of a subset of a topological space is defined with open subsets of the subspace topology, not with open sets of the bigger space.
fascinating video, when i first saw the proposed question i immediately thought about continuity and came to the conclusion that the curve is not connected, as the limit of the function at x=0 is undefined therefore the curve is not continuous at x=0. but the rest of the video really intrigued me as it gave me a new perspective on the idea of "connected"
Very nice and clear as usual. Just to nitpick, the sentence _"the sets contain at least part of their common boundary"_ *implies* that two sets touch, but it's not a *definition* since two sets can touch without having a common boundary, e.g. the sets of points at a distance of at most 1 and 2 respectively from the origin.
Because sine is a cyclical function you can substitute 0 for 1/0 at the origin. Or another way of looking at it is if you take the mean value of any 'path' of length 0, the minimum and maximum (and therefore the mean) will be the same. The 0 length path at the origin has a different minimum and maximum but as it covers the full range of the functions values, they are by definition the positive and negative extremes for that function and the mean value of 0 is correct.
As a hint for getting to the third definition I came up with an intermediate definition that is still somewhat intuitive. Instead of defining a boundary point you simply define a limit point of a set A as a point x ( either inside or outside A) in which every neighborhood of x contains some member of A. Then the closure of A is just A plus all of A's limit points (which, by definition, will include all the boundary points of A as well as interior points). Then the equivalent definition of "touching" between two sets A and B will be that either 1.) A contains a point in the closure of B, or 2.) B contains a point in the closure of A. The proof that this definition of "touching" is equivalent to the boundary-based definition shouldn't be too hard. This is the only missing link I have yet to think through. Then the next second step is to refine the definition of touching for the case where we just assume nothing exists outside of A and B. This is a little cleaner because limit points outside of A U B no longer exist. They are now outside of the domain, i.e. outside the universe we are considering. They become topologically analogous to infinite limits (kind of an aside, but interesting to think about). In any case, if we throw away all the limit points external to A U B, then the definition of "touching" reduces to either A or B having ANY external limit point ( since if a point is not in A, it must be in B, and vice verse ). At this point the only thing left to do is show that "both open and closed" is equivalent to "contains all of its limit points" within the new restricted universe where nothing exists outside A U B. Once you do that you have the third definition. I know I'm close because "contains all of its limit points" is a pretty popular alternate definition of closed in real analysis texts. This would be so much easier to show using illustrations but I don't have the talent of the author of this video unfortunately. :(
There's a fun puzzle related to this. Consider a (closed) square ABCD, with the corners labeled so that A is opposite C and B is opposite D. Find two disjoint subsets of the square, one containing A and C and one containing B and D, such that each subset is _connected_ (in the touching subsets sense). Hint: both subsets must be neither open nor closed.
@@mmmmmmmmmmmmmSearch "Counterintuitive topological result concerning connectedness" (including quotation marks). It should lead you to a post on Math Stack Exchange discussing this. There are some images in the comments
You could take the set of all points with rational coordinates excludind A and C, and the set of all points with both coordinates irrational, merged with A and C
I subscribed immediately 'cause of this awesome vid! As a physics student, it's been difficult for me to have an intuitive grasp of topological properties like this. I hope you do a video about COMPACTNESS!!!
There's always the "room" for rationals (b/c compact rows), and more that infinitely much space for real numbers. The boundaries got duplicated at 10:04. Before that, they we correctly shown belonging to only one set on cutting.
It's simple. We have a graph defined as f(x)=sin(1/x); at x=0 it has no value because decision by 0 is not possible. Therefore the curve is disconnected at a single vertex, exactly at x=0.
It's a slightly more advanced topic that I never fully got through, but another really interesting topic involving definitional motivation is the concept of dimensionality. Hausdorff dimension vs. topological dimension would have a discourse roughly analogous to path-wise connectedness vs topological connectedness. Hausdorff dimension is more complicated, but also more intuitive IMO. In the case of Hausdorff dimension, instead of the topologists sine curve, pathological cases where intuition fails are fractals! You'd think that the boundary of a 2 dimension object must be a 1 dimensional object, but the boundary of the Mandelbrot Set has Hausdorff dimension 2. In other cases you can get a fractional Hausdorff dimension ( hence the name "fractal" ).
Hi, the graph (sin(1/x)) is connected at zero because it is a normal graph just like (sin x). They are both normal graphs, I used Desmos to graph (sin(1/x). If we graph (sin x) and (sin(50x)) then we get 50 oscillations per unit of length than with the 1 oscillation with (sin x) .And we can continue up to (sin (infinity*x) which will get us an infinite number of oscillations per unit of length which is the same thing as (sin(1/x)). Thanks for your educational videos.
The function is clearly undefined at x = 0, but its behaviour infinitely close to zero is interesting. f(-𝛿) = -1 and f(𝛿) = 1 for 𝛿 -> 0, both sides approaching zero infinitely fast towards the origin. So in a sense, f(±x -> 0) maps to every value between -1 and 1 except 0.
In my head, I define connected shapes as shapes that, at all points, have a continuous link. For line and curves, each point must connect with exactly two points to count. More and that's seen as passing through itself at at least one point. For geometric shapes you define an inside and outside. Outside is defined as having access to infinity without touching inside. In the case of "hollow" shapes like a ring, inner "outsides" are considered their own shape and these follow their own rules for defining in/out, and their inside is considered outside for the larger shape. If both insides can continuously connect without breaching a border or the outside, they are connected. So if two points are seperated by the border of a shape, they cannot connect. By my definitions, this sine function is connected as long as no points from either half touch x=0 aside from the final "infinity" points of each wave. I should mention that I am, in fact, a gay dog on the internet with no qualifications in this field, so my definitions may be incomplete or flawed. This is just an intuition opinion born of boredom
I do not understand the _textbook definitions_ at the end. Could anyone briefly explain them? Especially the first one~ It states a set is connected when it is NOT the union of two non-empty disjoint open sets. The way I understand it is having a connected space means there is no way to ever separate it into a two-subset partition where both are open, contain at least one point each and have none in common. So a _non-connected_ set would HAVE a possible splitting blueprint that produces two such subsets. Let’s take the example of the set made up of two balls in the plane (filled-in circles [discs] in Euclidean geometry space), putting each ball a fair distance from each other (they don’t overlap, nor are they tangent to one another). Just the most basic 2D example ever. In that case… what’s the partition? Which 2 disjoint non-empty open pieces make that set up? And well… for the second one, the video already states (but doesn’t really explain) much of the non-obvious stuff about it.
I would argue that the path definition actually leads to Connected. The problem with the argument in the video was that it assumed we could only include 1 point in the center line on the path, while we can actually include multiple. A parametric function, as used here, is able to have different outputs with the same x-coordinate.
This was my first thought... but in order for this to work, a parametric function (x(t),y(t)) must be defined leading from some point (a,sin(1/a)) to (b,sin(1/b)), where a0, such that its speed sqrt(x'(t)^2 + y'(t)^2) is finite for all values of t∈[0,1]. I don't think that's possible, because the parametric function must traverse an infinite distance when it crosses at x=0.
@Morphocular regarding the addition of the points of the segment [-1,1] on the y axis to either the right side or the left side: I understand that when we add it, we end up having a connected set, but that set is not the set of points that are on the curve y=sin(1/x) (i.e. the graph of the curve). Again, the graph of sin(1/x) does _not_ contain the segment, so why are we adding points to it that don't belong to the set in the first place? As I understand you showed that an augmented set is connected but that doesn't mean that the original set was also connected. Could you elaborate on this please?
He is building a topological space that has the property that it is connected, but not arcwise-connected, to demonstrate that these notions are not equivalent. If you don't add any points with x coordinate 0, the set is disconnected, and therefore not interesting for the purpose of the video.
Hm. I rather prefer the following transformation instead of that reciprocal one presented here: x → arctanh(x), y → arctanh(y). So *f'(x) = tanh(f(arctanh(x))).* This way the entire infinite ℝ² is mapped on to the finite square [-1;1]×[-1;1], where ±∞ is mapped on to x=±1 and y=±1. And yes this is a bijective transformation given arctanh and tanh being bijective functions/transformations.
Why dont you make videos. Real analysis Complex analysis Linear algebra Modern Algebra, in a complete sense. Also on Topological space. Derivation of their theorem. An appeal to you. I really enjoy your way of explaining maths. Simple and sturdy. Priceless videos.
Before watching the video. Yes, the line connects both halves. The reason why is that at any point of the curve, lim x+, lim x- and x are equal. When lim x± is y=0 then we define continuity as fulfilled if the line contains x which it does.
Great video! Doesn't the second definition require a metric? Given that a ball is defined as points within a certain distance of another point. How could we define "connectedness" without relying on metrics?
It is possible to define "connectedness" without a metric, and it is based on the idea of an "open set" of a topological space (which doesn't have to be a metric space). A topological space is a set together with a collection of subsets satisfying some axioms which generalise how the open balls of R^n behave. We call these subsets "open sets", and in this general setting, we can define a topological space to be connected if it is not the disjoint union of two open sets.
iirr the issue with path continuity for the topologist's sine function isn't quite that "no value could work", it's that you don't have enough domain space in a regular function to reach the origin while going through all the oscilations. If you had a long function, i.e., one defined over the domain [0,+inf[ + [-inf,0] you can find a "long path" that connects the two halves for any number you choose as the connection.
[0;1] is isomorphic to the (-inf; +inf). (Their cardinality is the same aleph-1). We have "enough space". For example the set defined by function y(x) = x^2 * sin(1/x), y(0) = 0 is connected, even though the path between some points has an infinite length. The problem with the function from the video is that the path is "undefined" in x=0. Its limit doesn't exist, so it can't be continuous. Which contradicts with the definition of the path. Therefore path doesn't exist.
If you take the real projective line as your definition of infinity, then you have the option for a third argument: the Topologist's Sine Curve is path-connected *through infinity*, because it has the same limit in both the positive and negative directions: zero. This only works because the real projective line is even "neater" than the regular real number line that does not admit any infinity as a member. That's the "projective" coming into play; it treats the number line as the projection of a very large circle where the top is 0 and the bottom is unsigned infinity. This also works well to describe the limits of 1/x itself: taking x to 0 gives you infinity and taking x to infinity gives you 0, because 1/x is just turning the line inside out. If you work with signed infinities then taking x to 0 gives you two values based on *how* you get there, and this whole argument falls apart because infinity does not loop around.
I am proud of myself for instantly coming up with the definition of two points which have a path inside the shape Edit: I didnt say path, I said an infinite amount of points with have an infinitasimally small distance between them Does that change things?
10:34 yes. There is no continuous path connecting the two circles. However, there is a “path” of close points. An example of such “path” is rather simple. First, take a path that would connect the two points if the common point was present. Next, remove it. Now all points are still “infinitely close”
You're describing "locally connected" which is a little complicated to define, but basically: For any point in the set, you can always find an arbitrarily small connected open subset containing that point. The 2-open-circles set is not connected but is locally connected, and the topologist since curve is connected but not locally connected.
@@fullfungo but if one defines "infinitely small..." as the smallest possible distance between two points (meaning they are touching), that would exclude any gap, or? Since the boundaries doesn't belong to open sets, to share them (in case of two, side by side squares more realistically visualised on a pixel based screen) doesn't fulfill the requirement of smallest distance. Any analogy from physics for an open set would help me a lot...
@@Littleprinceleon two points can’t touch. At least not in any sensible way. If they are distance 0 apart, then they are the same point. If the are at a greater distance, then there are *a lot* of points in-between. If you want two points so that no other point can fit between them, then they must actually be the same points. The reason why the case with a gap is different is because we are using sequences of points “to the left” and “to the right”. None of these sequences has the “closest” point to the gap. But for any distance there are points that are closer.
Just wanted to provide an inital guess(im at the 1:43 mark). My guess is it is conected because it is continous at all points except x=0 so we only have to worry about the funtion being discontinous at 0. Its connected because for any distance from 0 N their will always be a point P on the function within N of 0. If we choose P to be (N, f(N)) we will have a point N away. And N can be arbitrarily small
Another observation i have made is that for any point on the funtion, f(x), the distance from the function to the line segment at 0 is x because the line segment will always have a point with the same y value as the function. This should hold as x goes to 0
pausing at 1:55 to take a crack at my guess: If you define the parametric, piecewise function f(t) = (t + 1, sin(1/(t+1))) for t < -1 (0, 1 - t) for -1
I like how 8:12 is one of the most commonly replayed parts of the video, because it's the part where people absently watching going 'huh? what? wait, run that by me again' and replay it LOL
The definition that comes to my mind is to say there are two ways to be "connected": 1)two points connected to the same point(for example Point A and Point C is connected to Point B), are connected (A and C are also connected) 2) A point that has another point within a circle the radius of which is 1/inf is connected to that point
Here's a version of it in more mathematical statements: "A set is connected if all points in it are connected" "Two points are connected if they are within a distance of 1/inf from each other OR they both share a connection to a given point"
The idea that the curve" wiggles" more and more as you approach zero can also be seen this way: What are the x-intercepts? solve sin(1/x)=0 means 1/x=0+k(pi) and so x = 1/(k(pi)). Now k is assumed integer and in this situation k cannot be zero. As k goes to (minus) infinity, you get x-intercepts closer and closer to the origin. But in between two consecutive x-intercepts, the curve still has to attain a maximum or minimum, that is 1 or -1. I think this address 5:12 pretty well.
When i do my homework and i think that math is boring i watch your Videos and then im thinking about becoming a mathematician. This stuff is very interesting
Another way to look at this is through fractals, such as space filling curves, as even though they look like they fill the space, if you zoom in infinitesimally, there's still some empty area
2:30 : My attemt: "A shape is connected, if, without entering their borders, its impossible to surround any of its supposed components with an infinitley thin line, while leaving others unencompassed." ---- wouldnt be mutch use in programing i guess¿?, and it only works if infinietly small lines work as i think they do, and i am no expert whatsoever Edit: and it only works in 2 dimensions i guess :(
I also agree that a disconcertingly high number of math textbooks (and teachers) skip over the motivation behind their definitions. There have been so many times where I've been reading a math textbook and I encounter a definition with a bunch of theorems that follow, and I just have to sit there and think... why??? Where does this definition come from? Why do we care about studying it?. It's a real problem, one that I hope future educators and textbook writers will try to fix.
I played around with Desmos and input other trig functions of 1/x, with similarly brain breaking results. I also input tan(x!) and part of the right side of the graph looks like a solid colored rectangle and inputting csc(x!) and sec(x!) look like a similarly solid rectangle with a square cut out of it.
I think it is connected. As you move toward the origin, y is always between 1 and -1 so no matter where it actually is, it will always touch the line at x=0. Of course the problem is that lim x->0 of sin(1/x) from the left or right doesn’t converge so it kind of doesn’t make sense to claim that it’s between -1 and 1.
Before seeing the definition, my definition of connected for a graph is that for any point there's another point arbitrarily close to it from both sides of it and my definition for connected in topology is that for any two points there's at least one conneted (this meaning of connected would be the definition from the graph) path that starts at the first point and ends at the second and doesn't have any points not in the set inside of it
What about the function y=sin(1/(x^2)) ? The curve will approach zero with exactly the same y value from left and right, so both ends have to touch in the center, right?
Here's a big hint as to how a topological space can be open AND closed: unicorns. What can you say about the real unicorns that really exist in our world - not just the concept of unicorns? Can you say they have one horn? Sure! Can you say they have four hooves? Of course? Can you say that they're reptiles and speak Dutch? You might say, "No, that's ridiculous," to which I say: prove it! Show me a unicorn that's not a reptile and can't speak Dutch! You can't! You know why? Because there are no unicorns! Therefore, every statement you make about them is true, because there's no counterexample! Now with that in mind: can you say that the boundary points of the coordinate plane are included in the coordinate plane? Can you say they're NOT included?
There is a slight technicality / edge case in your "touching" definition of connectedness. If you split a set A into A and the empty set, they don't share any boundary points.
hey man idk if you'll see this comment but i just wanted to say the "vibe" you have is great. you feel like a nice uncle or cool nephew. idk how to describe it.
I have a way to alter the path way to make it work for the topologist sine curve. If you have any two points, they can be connected by a sequence of points which are at most a distance of ε apart. The set is connected if you can choose any positive value for ε no matter how small.
Hi all! I wanted to quickly address a misconception that I've been seeing in a lot of comments. A number of you seem to be confused as to why a path is defined using a function, since that would seemingly exclude paths such as circles which fail the vertical line test. The thing to realize is a "path" here is not defined simply as a function that takes a real number input and then outputs another real number, but it outputs points in the space, which for a 2D space means entire (x,y) pairs. So a path in 2D space is a function that takes real number inputs in the range [0,1] and outputs (x,y) pairs, not just y-values. So for example, a full circular path can be defined as f(t) = (cos t, sin t) which describes a point moving counter-clockwise around a circle of radius 1 (if t ranges from 0 to 2pi). If you think of t as representing time, then this literally describes a point moving around a circle with speed 1.
I hope that helps!
It’s called a parametric function. That’s all.
@@vinny5004 Every function has a parameter. You're thinking of parametric equations
@@viliml2763 he said parametric though
@@viliml2763 actually, its not always possible to find a parameterization of multivariable functions. and indeed, when you parameterize something you define a "function" not an "equation".
so what you said doesnt abide by the actual definitions of parameterizations, at least the ones i have seen.
Ok so the topologists sine curve doesn't have a limit as it approaches zero, or I guess it is undefined. Which makes sense that therefore the curve is "disconnected" by the first definition. But what if we add that line segment stretching from (0,-1) to (0,1)? Does it then become connected? Let's say you picked the points on the curve (-1,sin-1) and (1,sin1) and let them be t = 0 and 1 respectively. As such when t = 0.5 then f should = (0,0) and there is some range of values of t that will return points on the curve with an x value of zero. Right? What have I missed?
my one semester of analysis has finally paid off in the form of understanding this youtube video and knowing the definition of connectedness
Hee! I did my analysis course after studying topology. The assignment grader hated be for solving problems in terms of (eg) 'coverings of open sets'.
Three cheers for soft analysis!
@@zapazap Lol I did both and to be honest I never even though of the analysis definition
@@micayahritchie7158 I"m a programmer and I do a lot of code refactoring -- rewriting expressions so that they are more elegant. I was doing this with the epsilon_felta d definition one day. To my surprise, it brought me to the thteshold of the more general topological definition.
The intuition guiding my steps was not mathematical at all, but (as far as I could tell) solely with making the case expression more elegant.
It amazed (and amazes) be to think: can concern for (and practice with) code refactoring, over the space of a few hours, replicate a fifty(?) plus year jump From Wierstrass to topology? Might he (who is attributed with the epsilon-felts definition), simply by obsessively 'polishing his code', have been led to considerations of general topology?
The thought is tantalizing.
What might it entail for research today?
Cheers! :)
^.^
ALL HAIL STEPHAN ABBOT FOR ADDING A TOPOLOGY CHAPTER!
The structure of this video is fantastic! Every time I thought something was confusing or needed an example, it was immediately followed by further explanation or an example. Anticipating how people are going to follow along is difficult, so I'm very impressed with how well this video manages to do that
Sin(1/x) is the maths equivalent of a black hole. It's just freaking awesome.
Its a good visual for what happens to light near a black hole. Any frequency of light gets redshifted to a wavelength impossible to detect even against the CMB
With the event horizon being the point at which any escaping light becomes too redshifted to possibly detect
@@alextaunton3099 what about sin(1/x^2)
*yes*
my favorite comment thread of all time
Exactly!! We need to know the motivation in arriving at a particular definition in the textbooks. This would make learning more interesting.
But I guess it's practically impossible. Like here in this beautiful video it takes roughly 22 minutes to explain the motivation behind just 2 definitions. That's cool but we've encountered hundreds, if not thousands of definitions/theorems in various branches of maths so far from school to master degree level. So I think it's obviously why our education system is the way it is. We don't have infinite time to explain every possible concept the way this guy or 3b1b explains!!!! It just isnt practical
@@randomdude9135 As a biologist I also learned a lot of concepts, but two decades after university I realize that at least half of them could be omitted at the benefit of delving into more depth regarding the more basic ideas. Even at the expense of being less specialized...
I would guess that in math to be able to connect between different branches is even more important either for practical or for theoretical creativity.
Maybe some specializations teaching and exploring the historical development of individual sciences strictly focusing on the cognitive processes necessary for a fuller understanding of the concepts
I am finishing my masters in physics and I must say this is the clearest explanation on this subject I've ever seen. Keep up!
The topologist's sine curve is a very cool topological space, and it is a very natural place to get into the nitty-gritty of what connectedness means in all its forms. I think you have done that very well here.
However, one point of criticism: I think it's wrong to entirely fail to mention around 5:30 the possibility that there may be *more* than one value t that gives f(t) = (0, y). It doesn't change the result, but it does slightly complicate the argument (you need to choose your t* carefully), so I can understand glossing over it. But not mentioning it at all is far beyond just glossing over it, and I personally think that falls into the territory of "incorrect". You have a note for the pedantic when talking about neighbourhoods at 8:25, that could've been a good fit here too.
I've certainly been known to err, but from what I can understand, I'm not sure I'd glossed over all that much? In the case of multiple possible t* values, I think the basic argument I used works if we take t* to be the minimum such t* (and I think you can justify via continuity of f that such a minimum exists). Then the argument proceeds as normal (though you might have to swap out the two-sided limits for left-handed limits). Is that the subtlety you were referring to?
@@morphocular Yeah, more or less. Continuity says the set of ts where f(t) has zero x-coordinate must be closed, the intermediate value theorem says it must be non-empty, and Heine-Borel (or something) tells you it is compact and therefore contains its minimum. If you pick t* to just be an arbitrary such t, rather than the minimum, then the limiting argument tells you nothing.
@@MasterHigure Gotcha. And thanks for the feedback! Figuring out where to draw the line at how much detail to skip is a never-ending battle for me, so I appreciate your input. To be honest, I hadn't considered that multiple t* values could be much of an issue until I realized (just today!) that there could be a whole interval of them!
@@morphocular There is no objective place to draw the line. Heck, I don't even know that it make sense to speak about *consistency* with where you draw the line, much less correctness. I have told you one place where I disagree with your line drawing, and one where I agree, but that doesn't really mean I'm right and you're wrong.
@@MasterHigure True, but I can always use more data.
Next topic: compactness. You can present different notions of compactness like you presented different notions of connectedness.
I just love the ingenuity in that definition! A set is compact if every open cover has a finite subcover. That is actually such a cool definition of a concept that is intuitively so easy.
Nice prediction!
I love how in the last moments of the video he showed a graphic of the epsilon delta definition of continuity that instantly made me understand it despite eluding me for years at this point. Excellent video if anyone's reading this.
this was such a beautiful video i HAD to leave a comment, like, and subscribe.
what i've found really beautiful in mathematics, is when we start coming up with definitions to seemingly answer an extreme case of an intuitive problem, the solution is just "whatever you so desire". it took me sometime to appreciate this beauty, because at first it feels like a slap in the face, like when you're in an adventure story to find the truth of the world, meets an all-knowing-being find what is the meaning of life and they ask back at you "what is your meaning of life?". and this video really encapsulates that feeling.
also that 1st textbook definition of "connected" really blew me away. it's almost like they used the aforementioned (two open disks touching at their boundary, a.k.a. a union of two non-empty disjoint open sets) to come up with a definition. fascinating.
Yes. Much of mathematics is about the language to represent concepts. There are definitions, and not knowing them makes it impossible to solve a problem that requires knowing it. But does not make you any more or less smart or intuitive. However, once one finds out and understands a definition, there are many important and useful things that result (e.g., theorems that can be applied to large classes of problems), and then “correct” and “incorrect” answers arise from the definitions (so that our language remains consistent). This last point is sometimes lost on students, who get angry and argue against definitions (they are arbitrary, but must be accepted, like we accept the meaning of words in spoken language… but with much more rigor in mathematics!). This is futile, “by definition,” so don’t do it, and instead focus on understanding the deep meaning of the definitions and their applications. Great video.
Even more so... I think that a sufficiently intelligent being less occupied with limitations of its individual existence would ask back:
What do you mean by "meaning" and why is it important for you?
Interesting video. And I can certainly appreciate the point you made at the end about the importance of definitions and the difficulties mathematicians sometimes have formulating good definitions. I recall from my studies of calculus that it took a while for mathematicians to derive the definitions of limit and derivative that we take for granted today. Before our current definition of limit was proposed nobody could define limits in terms that did not verge on the mystic.
There's an excellent YT video called something like: the lore of calculus
eyeopening and excellently narrated. The visuals and transitions were just perfect as well. Perfect video on the subject, thank you for the hard work it took to make this!
Absolutely fantastic video. Even though I encountered this example 5 years ago, I can honestly say this is the first time I actually truly capture the subtle difference between the two definitions.
Also, kudos on the definition-motivation plea at the end. Great work!
After watching the intro, here's my guess: to connect the two "ends" of the curve around 0, those individual ends must have well-defined limits. Even though the overall curve does not have a well-defined limit at 0, you could imagine each individual end having one. However, for this curve, each side of the curve near 0 is just as badly-behaved as 0 itself.
Pretty good summary of the path portion
21:12 A follow up video on this equivalence would be awesome.
For #1, see that if you view the set (say, the two (almost) touching open/closed disks) not as a subset of a larger space but as a space itself, it has no boundary (there are no points outside of it as it is everything), and furthermore every connected component has no boundary and so it is an open set. If it has multiple components then it is the union of disjoint open sets.
For #2, see that being both open and closed at the same time means having no boundary (contain all boundary while containing none of the boundary = boundary is nothing)
Great video! I especially like the bit at the end when you compare semantics.
I think math's greatest problem is that definitions usually throw away all context and motivation which makes them happen in the first place.
It's like trying to describe an algorithm in assembly/binary, yeah it's useful for execution, but beyond terrible for understanding the ideas at play.
I love point-set topology for how completely insane and seemingly contradictory it is despite being entirely consistent. You didn't even get into locally connected and how it's separated from connected.
Also, I don't think the textbook definition of connected is that hard to explain, since a closed set is defined to be the complement of an open set. So if a non-trivial set in the space is both open and closed, then it's a non-empty open set, whose complement is also non-empty and open.
15:57 My mind went immediately to the Touching Subsets definition of connectedness and concluded the Topologists' Sine Curve is connected, before you reminded me about the path-connected definition.
My attempt at coming up with a definition for "connectedness," as you mention at 2:30. I wanted to see how I'd do before I see the answer.
Define a Connector set as being constructed by the following process. Start with a well ordered set of m points with m>1. Call this the seed set A = {a_i| i from 1 to m inclusive}. The connector set is the union of all of the line segments connecting a_i to a_i+1 for i ranging from 1 to m-1.
Now suppose we want to assess if a region R is connected. The set R is connected iff for all r_1 and r_2 in R, there exists a connector set A containing r_1 and r_2 st A is contained within R.
Edit: Ah I now see this definition only works if R is of at least 2 dimensions everywhere. Tricky!
I would argue that the curve with the line along the y-axis is still connected with the path definition (Disclaimer: I am not a mathematician, so feel free to correct me if I'm misusing terminology; more notes at bottom). There are only two possible scenarios for the "endpoints" of the separated curves without the vertical line (these "endpoints" reflect the y-value of the function immediately to the left and immediately to the right of x=0). The endpoints are either located at effectively the same point to create what would normally be a removable discontinuity or located at different y-values to create what would normally be a jump discontinuity. These types of discontinuity usually refer to functions where the limits at the x-value are known. In this case, the limits from the left and right are undefined due to the oscillating pattern. However, the function is bounded between two y-values, so we know that the "endpoints" of both sides will each have a y-value between -1 and 1. This would mean that adding a line at x=0 that ranges from y=-1 to y=1 should attach to both disconnected "endpoints" regardless of where they are positioned relative to the y-axis. Although we cannot determine the precise values of the path function as it meets the y-axis, similar to the way that we cannot determine a finite limit at x=0, an indefinite path function must exist that traces the vertical line to connect the two segments.
Note: This is the result of a bunch of concepts and theorems back from Calculus 1 that have been scrambled together from memory to try to make something that resembles a decent explanation. The main inspiring concepts aside from discontinuity were the Intermediate Value Theorem and the Squeeze Theorem. It's also 2:30am as I'm finishing this (I got hyperfixated on number stuff again), so please let me know if there's something that doesn't make sense, and I'll try my best to clarify later
One of the best videos on connectedness i've seen so far, and i've seen a lot
Topologist's sine curve is one of my favourite so I watch this! If I don't misunderstand, the definition of the curve T you used is T:=the union of left part L, right part R and Y={(0,y): -1
So nice to see a video on point set topology. The _best_ topology. :)
Very interesting video. But please kindly clarify regarding the Ordered Square (17:41)
1. Why a space-filling curve, such as the (infinite iteration of the) Hilbert curve cannot be considered an ordered square?
2. If it is indeed possible to use the Hilbert curve as a 1D path through the ordered square, then would this also enable us to use the path idea to define connectedness?
Thanks in advance to anyone explaining these questions.
I'm not sure, but I think a space filling curve _could_ be used, but in topology, there's always a definition hiding somewhere of what topological space you're dealing with, and that includes a so-called topology. The ordered square is using the order topology for the given ordering, while using a space filling curve would give you a different topology (if even possible, idk), so you'd be dealing with a different topological space, just like how the ordered square is different from the regular square topologically, even though they are the same when only looking at what points there are
It has to be continuous in the order topology, not the usual Euclidean topology. The order topology is the topology where open intervals or any union of open intervals are considered open sets.
@@decare696 Thanks
@@stevenfallinge7149 Thanks
The Hilbert space filling curve preserves 'closeness' of points with respect to the standard square, but not to the ordered square.
Before I watch the rest of the video, you asked for my thoughts. I think the graph... is both connected and disconnected. On one hand, in the same way that there are infinitely many values between 0 and 1, now we find that no matter how far we zoom in on the y-axis, we can always zoom in further and insert a new point on either side that will have a different y-value. There is no meaningful "final" value of y for the "final" position x immediately below and above 0, so there is no well defined "attachment point" to which the y-axis can be "connected" with the curves on either side, hence disconnected. But, this oscillation continues infinitely, so there is no break or discontinuity between the y-axis and the curves, hence connected. Can limits help with this, maybe? As they say on "Hollywoo Stars and Celebrities: What do they know, do they know stuff? Let's find out," let's find out!
I have been studying civil engineering since last October and am struggling to grasp the concept of continuous functions. Thanks to your videos, some things have started to click and the gears are starting to move.
That may be as well because I have been studying for the upcoming math exams for the entirety of last week, but hey, i’ll take what I can get.
21:18 “But in any case, one of the things I hope you take away from this video is an appreciation of the ingenuity that goes into crafting a good mathematical definition. So often in Math, the focus is on the big theorems, proofs, and calculations, but I feel we don’t often pause to appreciate all the creativity and cleverness that go into some of the definitions. Maybe it’s partly because textbooks and lectures often just state definitions unmotivated before quickly moving on to the theorems and proofs. *But just as every theorem needs a proof, I think every definition needs a motivation.* And it’s an important mathematical skill to cultivate too! One that I worry too often goed unexercised. So the next time you encounter a strange mathematical definition out in the wild, take a minute to flex that mathematical muscle and try to understand what motivated it, and also appreciate the work that went into crafting it.”
"Just as every theorem needs a proof, every definition needs a motivation"
It feels like I'm the only one in my math class thinking this, including the teacher ;)
This is a lovely video. It took my right back to my first topology course. Thanks for sharing.
Brilliant concept, wonderful video. Great job.
I totally agree with you, each definition should be motivated before diving into the proof
The topologist' sine curve is defined as the closure of the set (x,sin(1/x)), 0
I'd say they don't agree because the topological definition doesn't include the requirement of a limit, only that the sets share a boundary which is in the whole set, which I interpret to mean the function has (at least one) y value for each x value.
Even in the 'path definition', if you remove the requirement that the limit exists, you can define a (discontinuous) function that works (but again that is not the definition of "path")
I'm taking topology this semester, so seeing this video made me very happy :)
Another question: instead of the topologist's sine curve, if we modify it such that we only add the point (0, 0) instead of a whole strip from (0, -1) to (0, 1), would the resulting set still be connected?
Yes, this is actually also how the topologists sine curve is usually defined.
It still has the same properties as the version in the video (which is the closed topologist sine curve) It's disconnected in some definition but connected in others
Yes, it would be. I think the proof is essentially the same.
It does not even have to be (0, 0). You can add any non-empty subset of the line segment to make it connected, as you just need to guarantee one point of the common boundary in either parts of a split.
No. It won't. There is still no continuous path that can connect them.
Edit: nvm
Really nice introduction to these concepts! However, an important thing I always try to teach my students is that topological notions are always relative. An Intervall in R may be open, however the same interrval in R^2 will not be w.r.t. the standard topology on R^2. Usually the topologist's sine curve is not just considered as a subset of R^2 with its standard topology, but rather as a topological space in it's own right, by giving it the subspace topology. In this case your argument, that all points of the curve are boundary points, is incorrect. That's also why connectedness of a subset of a topological space is defined with open subsets of the subspace topology, not with open sets of the bigger space.
Loved this video! I managed to guess the path definition but the subset definition was really cool.
You are one of the only math creators I can understand without getting bored
This sine function is used in Physics in what is called "asymptotic methods" usually in integrals and usually disguised as a complex exponential.
fascinating video, when i first saw the proposed question i immediately thought about continuity and came to the conclusion that the curve is not connected, as the limit of the function at x=0 is undefined therefore the curve is not continuous at x=0. but the rest of the video really intrigued me as it gave me a new perspective on the idea of "connected"
Such a good watch! Very interesting topic and a well made video. Cheers!
Very nice and clear as usual.
Just to nitpick, the sentence _"the sets contain at least part of their common boundary"_ *implies* that two sets touch, but it's not a *definition* since two sets can touch without having a common boundary, e.g. the sets of points at a distance of at most 1 and 2 respectively from the origin.
Because sine is a cyclical function you can substitute 0 for 1/0 at the origin. Or another way of looking at it is if you take the mean value of any 'path' of length 0, the minimum and maximum (and therefore the mean) will be the same. The 0 length path at the origin has a different minimum and maximum but as it covers the full range of the functions values, they are by definition the positive and negative extremes for that function and the mean value of 0 is correct.
As a hint for getting to the third definition I came up with an intermediate definition that is still somewhat intuitive. Instead of defining a boundary point you simply define a limit point of a set A as a point x ( either inside or outside A) in which every neighborhood of x contains some member of A. Then the closure of A is just A plus all of A's limit points (which, by definition, will include all the boundary points of A as well as interior points). Then the equivalent definition of "touching" between two sets A and B will be that either 1.) A contains a point in the closure of B, or 2.) B contains a point in the closure of A. The proof that this definition of "touching" is equivalent to the boundary-based definition shouldn't be too hard. This is the only missing link I have yet to think through.
Then the next second step is to refine the definition of touching for the case where we just assume nothing exists outside of A and B. This is a little cleaner because limit points outside of A U B no longer exist. They are now outside of the domain, i.e. outside the universe we are considering. They become topologically analogous to infinite limits (kind of an aside, but interesting to think about). In any case, if we throw away all the limit points external to A U B, then the definition of "touching" reduces to either A or B having ANY external limit point ( since if a point is not in A, it must be in B, and vice verse ).
At this point the only thing left to do is show that "both open and closed" is equivalent to "contains all of its limit points" within the new restricted universe where nothing exists outside A U B. Once you do that you have the third definition. I know I'm close because "contains all of its limit points" is a pretty popular alternate definition of closed in real analysis texts.
This would be so much easier to show using illustrations but I don't have the talent of the author of this video unfortunately. :(
There's a fun puzzle related to this.
Consider a (closed) square ABCD, with the corners labeled so that A is opposite C and B is opposite D.
Find two disjoint subsets of the square, one containing A and C and one containing B and D, such that each subset is _connected_ (in the touching subsets sense).
Hint: both subsets must be neither open nor closed.
What's the solution?
@@mmmmmmmmmmmmmSearch "Counterintuitive topological result concerning connectedness" (including quotation marks). It should lead you to a post on Math Stack Exchange discussing this. There are some images in the comments
@@columbus8myhw Thank you!
@@columbus8myhwi cant find it anymore 😭
You could take the set of all points with rational coordinates excludind A and C, and the set of all points with both coordinates irrational, merged with A and C
What a surprisingly cleanly-explained video
I subscribed immediately 'cause of this awesome vid! As a physics student, it's been difficult for me to have an intuitive grasp of topological properties like this. I hope you do a video about COMPACTNESS!!!
18:12 "this picture is a lie". Love it!
Awesome explanation! keep going!! looking forward for more such amazing Math Content
There's always the "room" for rationals (b/c compact rows), and more that infinitely much space for real numbers.
The boundaries got duplicated at 10:04. Before that, they we correctly shown belonging to only one set on cutting.
To which subset will the boundary belong to?
@@Littleprinceleon if you cut existing "continuous" space, then only to one of them, I guess.
This channel needs a few million subs, So good
thanks for making the end of this video, you didn't have to include that boundary, but you did ❤
It's simple. We have a graph defined as f(x)=sin(1/x); at x=0 it has no value because decision by 0 is not possible. Therefore the curve is disconnected at a single vertex, exactly at x=0.
This is just fascinating and mysterious, and maddening to me, so entertaining, beats all tv programs, tbh
It's a slightly more advanced topic that I never fully got through, but another really interesting topic involving definitional motivation is the concept of dimensionality. Hausdorff dimension vs. topological dimension would have a discourse roughly analogous to path-wise connectedness vs topological connectedness. Hausdorff dimension is more complicated, but also more intuitive IMO. In the case of Hausdorff dimension, instead of the topologists sine curve, pathological cases where intuition fails are fractals! You'd think that the boundary of a 2 dimension object must be a 1 dimensional object, but the boundary of the Mandelbrot Set has Hausdorff dimension 2. In other cases you can get a fractional Hausdorff dimension ( hence the name "fractal" ).
These are extremely excellent videos. Hope you do way more especially about the weirdest math topics! Thank you
Hi, the graph (sin(1/x)) is connected at zero because it is a normal graph just like (sin x). They are both normal graphs, I used Desmos to graph (sin(1/x). If we graph (sin x) and (sin(50x)) then we get 50 oscillations per unit of length than with the 1 oscillation with (sin x) .And we can continue up to (sin (infinity*x) which will get us an infinite number of oscillations per unit of length which is the same thing as (sin(1/x)). Thanks for your educational videos.
short answer:
lim x->0 (x) => infinity => infinite oscillations
remediable gap in definition at x = 0
Very nice video. I hope you'll soon get a lot more subscribers.
The function is clearly undefined at x = 0, but its behaviour infinitely close to zero is interesting. f(-𝛿) = -1 and f(𝛿) = 1 for 𝛿 -> 0, both sides approaching zero infinitely fast towards the origin. So in a sense, f(±x -> 0) maps to every value between -1 and 1 except 0.
You can change it to be f(x)=sin(1/x) for x≠0 and f(0)=0, then it would be defined and continuous everywhere
Whoops nevermind, that for xsin(1/x), not this function. I got confused. Nevermind
In my head, I define connected shapes as shapes that, at all points, have a continuous link. For line and curves, each point must connect with exactly two points to count. More and that's seen as passing through itself at at least one point.
For geometric shapes you define an inside and outside. Outside is defined as having access to infinity without touching inside. In the case of "hollow" shapes like a ring, inner "outsides" are considered their own shape and these follow their own rules for defining in/out, and their inside is considered outside for the larger shape.
If both insides can continuously connect without breaching a border or the outside, they are connected. So if two points are seperated by the border of a shape, they cannot connect.
By my definitions, this sine function is connected as long as no points from either half touch x=0 aside from the final "infinity" points of each wave.
I should mention that I am, in fact, a gay dog on the internet with no qualifications in this field, so my definitions may be incomplete or flawed. This is just an intuition opinion born of boredom
I do not understand the _textbook definitions_ at the end. Could anyone briefly explain them? Especially the first one~
It states a set is connected when it is NOT the union of two non-empty disjoint open sets. The way I understand it is having a connected space means there is no way to ever separate it into a two-subset partition where both are open, contain at least one point each and have none in common. So a _non-connected_ set would HAVE a possible splitting blueprint that produces two such subsets. Let’s take the example of the set made up of two balls in the plane (filled-in circles [discs] in Euclidean geometry space), putting each ball a fair distance from each other (they don’t overlap, nor are they tangent to one another). Just the most basic 2D example ever. In that case… what’s the partition? Which 2 disjoint non-empty open pieces make that set up?
And well… for the second one, the video already states (but doesn’t really explain) much of the non-obvious stuff about it.
Very well-made!!! Thank your for your beautiful piece of explanation!! 😻😻
The "textbook definition 1" is pretty clearly equivalent to the one you explained imo.
I would argue that the path definition actually leads to Connected. The problem with the argument in the video was that it assumed we could only include 1 point in the center line on the path, while we can actually include multiple. A parametric function, as used here, is able to have different outputs with the same x-coordinate.
This was my first thought... but in order for this to work, a parametric function (x(t),y(t)) must be defined leading from some point (a,sin(1/a)) to (b,sin(1/b)), where a0, such that its speed sqrt(x'(t)^2 + y'(t)^2) is finite for all values of t∈[0,1]. I don't think that's possible, because the parametric function must traverse an infinite distance when it crosses at x=0.
This was actually a really good video wow, never thought I'd waych it for 20minutes, thank you s lot ^^
@Morphocular regarding the addition of the points of the segment [-1,1] on the y axis to either the right side or the left side: I understand that when we add it, we end up having a connected set, but that set is not the set of points that are on the curve y=sin(1/x) (i.e. the graph of the curve). Again, the graph of sin(1/x) does _not_ contain the segment, so why are we adding points to it that don't belong to the set in the first place? As I understand you showed that an augmented set is connected but that doesn't mean that the original set was also connected. Could you elaborate on this please?
He is building a topological space that has the property that it is connected, but not arcwise-connected, to demonstrate that these notions are not equivalent. If you don't add any points with x coordinate 0, the set is disconnected, and therefore not interesting for the purpose of the video.
As soon as you brought the definition of connectedness back to the problem, I went "ahhh... Divergent".
Hm. I rather prefer the following transformation instead of that reciprocal one presented here:
x → arctanh(x),
y → arctanh(y).
So *f'(x) = tanh(f(arctanh(x))).*
This way the entire infinite ℝ² is mapped on to the finite square [-1;1]×[-1;1], where ±∞ is mapped on to x=±1 and y=±1.
And yes this is a bijective transformation given arctanh and tanh being bijective functions/transformations.
Why dont you make videos.
Real analysis
Complex analysis
Linear algebra
Modern Algebra,
in a complete sense.
Also on Topological space.
Derivation of their theorem.
An appeal to you.
I really enjoy your way of explaining maths.
Simple and sturdy.
Priceless videos.
Before watching the video. Yes, the line connects both halves.
The reason why is that at any point of the curve, lim x+, lim x- and x are equal. When lim x± is y=0 then we define continuity as fulfilled if the line contains x which it does.
Great video! Doesn't the second definition require a metric? Given that a ball is defined as points within a certain distance of another point. How could we define "connectedness" without relying on metrics?
It is possible to define "connectedness" without a metric, and it is based on the idea of an "open set" of a topological space (which doesn't have to be a metric space). A topological space is a set together with a collection of subsets satisfying some axioms which generalise how the open balls of R^n behave. We call these subsets "open sets", and in this general setting, we can define a topological space to be connected if it is not the disjoint union of two open sets.
iirr the issue with path continuity for the topologist's sine function isn't quite that "no value could work", it's that you don't have enough domain space in a regular function to reach the origin while going through all the oscilations. If you had a long function, i.e., one defined over the domain [0,+inf[ + [-inf,0] you can find a "long path" that connects the two halves for any number you choose as the connection.
[0;1] is isomorphic to the (-inf; +inf). (Their cardinality is the same aleph-1). We have "enough space". For example the set defined by function y(x) = x^2 * sin(1/x), y(0) = 0 is connected, even though the path between some points has an infinite length. The problem with the function from the video is that the path is "undefined" in x=0. Its limit doesn't exist, so it can't be continuous. Which contradicts with the definition of the path. Therefore path doesn't exist.
This was a really well made video
Fantastic !! as usual for your standard !!
If you take the real projective line as your definition of infinity, then you have the option for a third argument: the Topologist's Sine Curve is path-connected *through infinity*, because it has the same limit in both the positive and negative directions: zero.
This only works because the real projective line is even "neater" than the regular real number line that does not admit any infinity as a member. That's the "projective" coming into play; it treats the number line as the projection of a very large circle where the top is 0 and the bottom is unsigned infinity. This also works well to describe the limits of 1/x itself: taking x to 0 gives you infinity and taking x to infinity gives you 0, because 1/x is just turning the line inside out. If you work with signed infinities then taking x to 0 gives you two values based on *how* you get there, and this whole argument falls apart because infinity does not loop around.
I am proud of myself for instantly coming up with the definition of two points which have a path inside the shape
Edit: I didnt say path, I said an infinite amount of points with have an infinitasimally small distance between them
Does that change things?
10:34 yes. There is no continuous path connecting the two circles.
However, there is a “path” of close points.
An example of such “path” is rather simple. First, take a path that would connect the two points if the common point was present. Next, remove it. Now all points are still “infinitely close”
@@fullfungo yeah I see
You're describing "locally connected" which is a little complicated to define, but basically: For any point in the set, you can always find an arbitrarily small connected open subset containing that point. The 2-open-circles set is not connected but is locally connected, and the topologist since curve is connected but not locally connected.
@@fullfungo but if one defines "infinitely small..." as the smallest possible distance between two points (meaning they are touching), that would exclude any gap, or?
Since the boundaries doesn't belong to open sets, to share them (in case of two, side by side squares more realistically visualised on a pixel based screen) doesn't fulfill the requirement of smallest distance.
Any analogy from physics for an open set would help me a lot...
@@Littleprinceleon two points can’t touch. At least not in any sensible way.
If they are distance 0 apart, then they are the same point.
If the are at a greater distance, then there are *a lot* of points in-between.
If you want two points so that no other point can fit between them, then they must actually be the same points.
The reason why the case with a gap is different is because we are using sequences of points “to the left” and “to the right”. None of these sequences has the “closest” point to the gap.
But for any distance there are points that are closer.
Just wanted to provide an inital guess(im at the 1:43 mark). My guess is it is conected because it is continous at all points except x=0 so we only have to worry about the funtion being discontinous at 0. Its connected because for any distance from 0 N their will always be a point P on the function within N of 0. If we choose P to be (N, f(N)) we will have a point N away. And N can be arbitrarily small
Another observation i have made is that for any point on the funtion, f(x), the distance from the function to the line segment at 0 is x because the line segment will always have a point with the same y value as the function. This should hold as x goes to 0
pausing at 1:55 to take a crack at my guess:
If you define the parametric, piecewise function f(t) =
(t + 1, sin(1/(t+1))) for t < -1
(0, 1 - t) for -1
I like how 8:12 is one of the most commonly replayed parts of the video, because it's the part where people absently watching going 'huh? what? wait, run that by me again' and replay it LOL
The definition that comes to my mind is to say there are two ways to be "connected":
1)two points connected to the same point(for example Point A and Point C is connected to Point B), are connected (A and C are also connected)
2) A point that has another point within a circle the radius of which is 1/inf is connected to that point
Here's a version of it in more mathematical statements:
"A set is connected if all points in it are connected"
"Two points are connected if they are within a distance of 1/inf from each other OR they both share a connection to a given point"
The idea that the curve" wiggles" more and more as you approach zero can also be seen this way: What are the x-intercepts? solve sin(1/x)=0 means 1/x=0+k(pi) and so x = 1/(k(pi)). Now k is assumed integer and in this situation k cannot be zero. As k goes to (minus) infinity, you get x-intercepts closer and closer to the origin. But in between two consecutive x-intercepts, the curve still has to attain a maximum or minimum, that is 1 or -1. I think this address 5:12 pretty well.
Masterpiece of a video
When i do my homework and i think that math is boring i watch your Videos and then im thinking about becoming a mathematician. This stuff is very interesting
Another way to look at this is through fractals, such as space filling curves, as even though they look like they fill the space, if you zoom in infinitesimally, there's still some empty area
The collision that never happens. Zeno darts towards his brother Zeno* as Zeno* run towards his brother Zeno. Both are forever unharmed.
2:30 : My attemt:
"A shape is connected, if, without entering their borders, its impossible to surround any of its supposed components with an infinitley thin line, while leaving others unencompassed."
----
wouldnt be mutch use in programing i guess¿?, and it only works if infinietly small lines work as i think they do, and i am no expert whatsoever
Edit: and it only works in 2 dimensions i guess :(
I also agree that a disconcertingly high number of math textbooks (and teachers) skip over the motivation behind their definitions. There have been so many times where I've been reading a math textbook and I encounter a definition with a bunch of theorems that follow, and I just have to sit there and think... why??? Where does this definition come from? Why do we care about studying it?. It's a real problem, one that I hope future educators and textbook writers will try to fix.
I played around with Desmos and input other trig functions of 1/x, with similarly brain breaking results. I also input tan(x!) and part of the right side of the graph looks like a solid colored rectangle and inputting csc(x!) and sec(x!) look like a similarly solid rectangle with a square cut out of it.
I think it is connected. As you move toward the origin, y is always between 1 and -1 so no matter where it actually is, it will always touch the line at x=0. Of course the problem is that lim x->0 of sin(1/x) from the left or right doesn’t converge so it kind of doesn’t make sense to claim that it’s between -1 and 1.
Before seeing the definition, my definition of connected for a graph is that for any point there's another point arbitrarily close to it from both sides of it and my definition for connected in topology is that for any two points there's at least one conneted (this meaning of connected would be the definition from the graph) path that starts at the first point and ends at the second and doesn't have any points not in the set inside of it
What about the function y=sin(1/(x^2)) ?
The curve will approach zero with exactly the same y value from left and right, so both ends have to touch in the center, right?
Here's a big hint as to how a topological space can be open AND closed: unicorns. What can you say about the real unicorns that really exist in our world - not just the concept of unicorns? Can you say they have one horn? Sure! Can you say they have four hooves? Of course? Can you say that they're reptiles and speak Dutch?
You might say, "No, that's ridiculous," to which I say: prove it! Show me a unicorn that's not a reptile and can't speak Dutch! You can't! You know why? Because there are no unicorns! Therefore, every statement you make about them is true, because there's no counterexample!
Now with that in mind: can you say that the boundary points of the coordinate plane are included in the coordinate plane? Can you say they're NOT included?
There is a slight technicality / edge case in your "touching" definition of connectedness. If you split a set A into A and the empty set, they don't share any boundary points.
16:12 Holy shit! That philosophy hit me deeper than it should
hey man idk if you'll see this comment but i just wanted to say the "vibe" you have is great. you feel like a nice uncle or cool nephew. idk how to describe it.
Thanks! :) I try to make each video feel friendly and inviting.
@@morphocular then you've certainly achieved that. keep it going man you're a really amazing content creator
Just a truly great video!
I have a way to alter the path way to make it work for the topologist sine curve. If you have any two points, they can be connected by a sequence of points which are at most a distance of ε apart. The set is connected if you can choose any positive value for ε no matter how small.
That's a good one. I'm pretty sure it's equivalent to the subset definition.
"sin(1/x), are you a connected curve?"
sin(1/x): "haha, essential singularity go brrr"
A function similar to this was in my first exam in college, mathematical analysis.
It was f(x) = { 0 if x=0; max(0, x^2 sin (1/x)) if x!=0}