The definition of the open set in the construction of the standard topology on Rn is isomorphic to the metric definition of openness in *Euclidean* metric spaces (i.e spaces with the euclidean/pythagorean metric), but the difference to realize is that definition of the open set in the construction of the standard topology doesn't rely on any distance metric per se but instead an isomorphic condition just using the algebraic structure of R (the "standardness" motivated by the apparent euclidean nature of real space). Further, besides independence, the other important point to realize is you can construct metric topologies from other metrics as well.
You claim that we can map topological space from abstract idea to be the space in real space (for any dimensions) by using the idea of how points connect to others. I also have a question if I would like a 1-d space "circle". How can I set a topological space? As you know in circle let's say there are 10 points in this circle. Point 1, 2, 3,..., 10 respectively. Point 1 has to connect with point 2 and point 2 must connect with point 3 and so on. That means in the set 1 must go with 2 and 2 must go with 3 but 1 must not connect with 3. How that's possible?
Why instead of a grid analogy, don't you use an infinitesimal bicycle tire with infinitely many spokes to connect a smooth tire to the axle? And then every point at the end of the axle is also itself an axle?
the best video i've ever seen, anywhere
Even for a non native English speaker like me, this was a very comprehensive introduction to the subject! Thanks!
'We're only interested in this because it mirrors the real world'. How refreshingly Hardy-esque!
Best video on the topic.
Just AWESOME!!
Can we say that usual topology and metric topologies are same things?
The definition of the open set in the construction of the standard topology on Rn is isomorphic to the metric definition of openness in *Euclidean* metric spaces (i.e spaces with the euclidean/pythagorean metric), but the difference to realize is that definition of the open set in the construction of the standard topology doesn't rely on any distance metric per se but instead an isomorphic condition just using the algebraic structure of R (the "standardness" motivated by the apparent euclidean nature of real space). Further, besides independence, the other important point to realize is you can construct metric topologies from other metrics as well.
does the set X really has to be an element of the topology??
You claim that we can map topological space from abstract idea to be the space in real space (for any dimensions) by using the idea of how points connect to others. I also have a question if I would like a 1-d space "circle". How can I set a topological space? As you know in circle let's say there are 10 points in this circle. Point 1, 2, 3,..., 10 respectively. Point 1 has to connect with point 2 and point 2 must connect with point 3 and so on. That means in the set 1 must go with 2 and 2 must go with 3 but 1 must not connect with 3. How that's possible?
{1,2} automatically means that 1 is not connected with 3
Why instead of a grid analogy, don't you use an infinitesimal bicycle tire with infinitely many spokes to connect a smooth tire to the axle? And then every point at the end of the axle is also itself an axle?