What is a Topological Space? 

The fundamental law of math: If you think you understand a topic, there's always a more difficult generalisation of that topic you don't understand (At least for me) But thanks for the link to the textbook, now I know what I'm doing this summer holiday (in Aus)

What an articulate and beautiful presentation

This OOP example really hit, thanks a lot

this was very clear and helpful, thank you!

Thanks. The video is quick but informative, it gives a nice overview.

good video; however, not all vector spaces are topological spaces, but they can all be given a topology. Same for metric spaces, but they have a nice induced topology.

Well done! I understood everything on the first try, love it!

Many thanks for the quick informative overview. I have following remark: ​​because a scalar product induces a norm, the shown diagramm can be extended with one more cycle.

“space” often but not always means topological space: measurable spaces, for example (I’d include vector spaces too but ok). another way of presenting this info that I’ve liked before is to talk about the “spatial” type structures that are on R^3. then you can talk about topological, affine, metric, vectorial, etc structures.

Please correct me if I'm wrong, but I don't think all vector spaces are metric spaces. Only normed vector spaces.

very great explanation! Thank you for that content

Thank you so much. It made it click right away. :D

you're diagram is quite confusing as if you are saying that every vector space is a metric space which is false.

Great video! Could you do one about generators of sigma algebras too?

Great job - very clear

what an amazing video. keep doing it. thank u always 💛

you clarified a lot for me. thanks a lott

Where are you? Keep it up, you are the best

My attitude may be too nitpicky for the general audience you are targeting with this video, but I don't think a vector space, as usually defined from the axioms, is a good example of a space to start with. It is more like an algebraic structure, it doesn't in itself have space-like features like distance or nearness. Now a k-vector space over a specific field k (like the real or complex numbers) may inherit such space-like features from its underlying field (a space-like real line or Argand plane). But the mathematical properties of a vector space itself (closed under linear combinations, spanning sets, choice of basis, matrix representations) aren't really space-like. The so-called vector "space" is more of an algebra-like coordinate system that you layer on top of some underlying genuinely space-like thing (such as the space that is the product of two or three non-parallel real lines). Admittedly, most calculus students will identify the algebra-like coordinate system with its underlying metric space with the Euclidean notion of distance, but I think that can be a source of confusion since such students would also be exposed to matrix-based linear algebra and programming over coordinate systems. I'm not a mathematician, my background is in software engineering, but I think it is better to think of the vector space as a coordinate system, a discrete thing with algebra-like operations, distinct from an underlying space or product space which is what has space-like features such as distance and continuous functions. Importantly, a vector space isn't a special kind of topological space (so the analogy with object-oriented programming and animals breaks down), only certain vector spaces over a space-like ground field "inherit" the features and mathematical properties of a topological space. A vector space over a finite field (such as a number system of modular integer arithmetic, like clock arithmetic modulo 12 or 24) is a perfectly legitimate vector "space" that isn't space-like, it is very discrete. Think of the discrete vector space generated by two digital clocks, one giving 12 hour numbers, the other giving military-style 24 hour numbers. This is pretty intuitive, you can program it just like other vector spaces, and do all the standard linear algebra like things including Gaussian elimination of matrix rows, but it is not a special kind of topological space or metric space. You can't do calculus over it or have topology-like continuous deformations. (Technically, I guess you can define a discrete topology over the (power set of the) set of number pairs, but that is structure beyond the vector space itself, and doesn't really support intuitions about spaces, it is one of those notorious mind-twisting abstract topology examples, a degenerate discrete topological structure).

I am not sure about this video. You started comparing to subsets of animals that they inherit everything from the bigger set. Then you said vectors, which are a subset of metric space have more properties then Metric space.

Excellent video

Very clear Thank You

@4:45 why is intersections considered for a finite number of sets but union for an infinite number of sets?

Nice explanation

- Excellent. - Thx.

Very Clear!

Thank you It was useful

Thanks fam.

MAD RESPECT.

Great video. Only suggestion is to buy a mic.

How could a union of subsets in tao not itself be in tao?

Nice overview

A general vector space is NOT a metric space!! There is no canonical metric on it!

I will to this day never understand the need, other than as a mere interesting curiosity, of putting a topology onto uncountably infinite groups, such as Lie Groups.

topology and abstract math is like opening the elder scroll from skyrim wtf is this shit lmao

Good

This video confused me more. I'm trying to be very precise when I learn things, and as mentioned, a topological space is not entirely a generalization of metric/vector spaces. Also, it would be more clear with the definition to say "if an arbitrary set of elements are in tau, then their union is also in tau." You simply said the latter piece, "arbitrary unions are in tau," without first stating that the sets that form the union are also in tau. You also refer to the pieces of the definition (5:29) as "the first axiom." These are not axioms, they are pieces of the definition. Being precise with language is important in math even if "just to give intuition." Also, need to give more pauses between sentences so we can absorb what you said.

*object* : that which you can imagine or draw; that which has shape

Mnmnn😊