Einstein never nearly understood TIME, E=MC2, F=ma, gravity, or ELECTROMAGNETISM/energy. He was, in fact, a total weasel. c2 represents a dimension ON BALANCE, as E=MC2 IS F=ma in accordance with the following: UNDERSTANDING THE ULTIMATE, BALANCED, TOP DOWN, AND CLEAR MATHEMATICAL UNIFICATION OF ELECTROMAGNETISM/energy AND gravity, AS E=MC2 IS CLEARLY F=ma: The stars AND PLANETS are POINTS in the night sky. E=MC2 IS F=ma, AS this proves the term c4 from Einstein's field equations. SO, ON BALANCE, this proves the fourth dimension. ELECTROMAGNETISM/energy is gravity. Gravity IS ELECTROMAGNETISM/energy !!! TIME is NECESSARILY possible/potential AND actual IN BALANCE, AS E=MC2 IS F=ma; AS ELECTROMAGNETISM/energy is gravity. INDEED, TIME dilation ULTIMATELY proves ON BALANCE that E=MC2 IS F=ma; AS ELECTROMAGNETISM/energy is gravity. Gravity IS ELECTROMAGNETISM/energy. Gravity AND ELECTROMAGNETISM/energy are linked AND BALANCED opposites, AS E=MC2 IS F=ma; AS ELECTROMAGNETISM/energy is gravity; AS gravity/acceleration involves BALANCED inertia/INERTIAL RESISTANCE; AS GRAVITATIONAL force/ENERGY IS proportional to (or BALANCED with/as) inertia/INERTIAL RESISTANCE. Gravity IS ELECTROMAGNETISM/energy. E=mC2 IS CLEARLY F=ma. This NECESSARILY represents, INVOLVES, AND DESCRIBES what is possible/potential AND actual IN BALANCE, AS ELECTROMAGNETISM/energy is gravity. Gravity IS ELECTROMAGNETISM/energy !!! By Frank DiMeglio
Around 15:40, what is written at the end of the Prop-Defn of _continuous_, should be "f-inverse(v) := {x in X | f(x) in V} is open in X" rather than open in Y. The diagram and spoken explanation are clear, just the written definition is inconsistent with them. A very nice set of videos, by the way.
It deserves more views, its overall easy to understand, even explained the motivation of it well. It explains the reason for intersection being finite, which i also have a hard time understanding where it comes up with.
Introductions to topology rarely explain the contravariance of continuous maps. Point-set topology is a study of proximity, but topology is also all about open sets, right? Well, then why aren't continuous functions required to map open sets to open sets? Why does openness of a set in a continuous function's range necessitate openness of the set's inverse image in the domain, but not the other way around? The key to explaining this inverted relationship might be a clearer description of the purpose of open sets. As Dr. Chan mentions in the video, open sets are a generalized abstraction of distance. That is, ordinarily distances are non-negative real numbers, but from the perspective of topology, each open set can be considered a non-zero "distance" in its own right. Two points x and y in open set A are up to A-apart, and a point x in A and a point y outside A are at least A-apart (or they're separated by A anyway). The key detail to observe here is that open sets represent _apartness_ or _separation_ , not proximity. The purpose of open sets is to distinguish points and patches of space, not to bring them together. Topology's contravariant preservation of openness, then, is nothing more than a modus-tollens-like consequence of its covariant preservation of proximity.
Hi! I like this video series very much but always has trouble with statement about pointwise convergence in metric spaces what can go wrong? Do you mean Dini's theorem? Maybe I'm missing something obvious.
During the introduction, you say that without additional geometric structure, it doesn't make sense to talk about functions going from one set to another but we can have functions going from one set to another without structure in any set. So, what is the significance of having the geometric structure?
Daniel is talking of continuous functions. Functions can make sense without structure but a notion of continuity is hard to define without further structure.
this is the best version of teaching topological sapces I have ever seen! You are spectacular I finally got this concept thanks to you!!
Einstein never nearly understood TIME, E=MC2, F=ma, gravity, or ELECTROMAGNETISM/energy.
He was, in fact, a total weasel.
c2 represents a dimension ON BALANCE, as E=MC2 IS F=ma in accordance with the following:
UNDERSTANDING THE ULTIMATE, BALANCED, TOP DOWN, AND CLEAR MATHEMATICAL UNIFICATION OF ELECTROMAGNETISM/energy AND gravity, AS E=MC2 IS CLEARLY F=ma:
The stars AND PLANETS are POINTS in the night sky. E=MC2 IS F=ma, AS this proves the term c4 from Einstein's field equations. SO, ON BALANCE, this proves the fourth dimension. ELECTROMAGNETISM/energy is gravity. Gravity IS ELECTROMAGNETISM/energy !!!
TIME is NECESSARILY possible/potential AND actual IN BALANCE, AS E=MC2 IS F=ma; AS ELECTROMAGNETISM/energy is gravity. INDEED, TIME dilation ULTIMATELY proves ON BALANCE that E=MC2 IS F=ma; AS ELECTROMAGNETISM/energy is gravity. Gravity IS ELECTROMAGNETISM/energy.
Gravity AND ELECTROMAGNETISM/energy are linked AND BALANCED opposites, AS E=MC2 IS F=ma; AS ELECTROMAGNETISM/energy is gravity; AS gravity/acceleration involves BALANCED inertia/INERTIAL RESISTANCE; AS GRAVITATIONAL force/ENERGY IS proportional to (or BALANCED with/as) inertia/INERTIAL RESISTANCE. Gravity IS ELECTROMAGNETISM/energy.
E=mC2 IS CLEARLY F=ma. This NECESSARILY represents, INVOLVES, AND DESCRIBES what is possible/potential AND actual IN BALANCE, AS ELECTROMAGNETISM/energy is gravity. Gravity IS ELECTROMAGNETISM/energy !!!
By Frank DiMeglio
You're extremely talented explaining mathematical concepts in a logical, understandable, engaging way! Thanks a lot!
The first video gives me the big picture of point set topology. Great!
Around 15:40, what is written at the end of the Prop-Defn of _continuous_, should be "f-inverse(v) := {x in X | f(x) in V} is open in X" rather than open in Y. The diagram and spoken explanation are clear, just the written definition is inconsistent with them.
A very nice set of videos, by the way.
Very good introductory video.
Great video
It deserves more views, its overall easy to understand, even explained the motivation of it well. It explains the reason for intersection being finite, which i also have a hard time understanding where it comes up with.
Introductions to topology rarely explain the contravariance of continuous maps. Point-set topology is a study of proximity, but topology is also all about open sets, right? Well, then why aren't continuous functions required to map open sets to open sets? Why does openness of a set in a continuous function's range necessitate openness of the set's inverse image in the domain, but not the other way around?
The key to explaining this inverted relationship might be a clearer description of the purpose of open sets. As Dr. Chan mentions in the video, open sets are a generalized abstraction of distance. That is, ordinarily distances are non-negative real numbers, but from the perspective of topology, each open set can be considered a non-zero "distance" in its own right. Two points x and y in open set A are up to A-apart, and a point x in A and a point y outside A are at least A-apart (or they're separated by A anyway).
The key detail to observe here is that open sets represent _apartness_ or _separation_ , not proximity. The purpose of open sets is to distinguish points and patches of space, not to bring them together. Topology's contravariant preservation of openness, then, is nothing more than a modus-tollens-like consequence of its covariant preservation of proximity.
Good comment
Hi! I like this video series very much but always has trouble with statement about pointwise convergence in metric spaces what can go wrong? Do you mean Dini's theorem? Maybe I'm missing something obvious.
I hope you continue
Thank you for your lectures!
Please, can you help with a video on topological semigroups I understand and love your teaching.
i like this
Question: in the definition of continuous function you have for every open V ⊆ Y, f^-1(V) is open in Y. Shouldn't f^-1(V) be open in X?
excellent thank you!
During the introduction, you say that without additional geometric structure, it doesn't make sense to talk about functions going from one set to another but we can have functions going from one set to another without structure in any set. So, what is the significance of having the geometric structure?
Daniel is talking of continuous functions. Functions can make sense without structure but a notion of continuity is hard to define without further structure.
Having geometric structure can help us understand point set topology, bc geometry is more concrete than sets