How to think/see point-set topology abstractly?

Question

I've started learning point-set topology this semester. I've learned basic material about:

• topology on a set
• topological space
• open sets
• closed sets
• clopen sets
• closure
• neighborhoods
• interior point
• interior
• exterior
• boundary
• boundary point

However, whenever I think of terms such as neighborhood, interior point, boundary $\dots$, I tend to think of them in terms of $\mathbb R^2$. (e.g., circles on the plane)

Is there a way for me to think of these, and future topology terms, abstractly?Intuitively?

What do you see in your mind, or think about, when you hear these terms? How do you look at this?

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I like to give this example from abstract algebra for what I mean when I say intuitively:

If we have a group and we mod out by the commutator subgroup, what's basically going on is that we are setting all the elements that do not commute equal to the identity and thus we are left with a quotient group where everything commutes.

Answer 1

The mathematical expression used in your question allows me to happily point you to following:

Point-Set Topology / mathworld.wolfram.com

Studies in this area create a foundation from which other branches of math can be built.

Just think of how far we've traversed from Euclid to modern math in exploring the concept of

Point (geometry).

Now, working in the set-theoretic framework, we can boldly regard an element of a set as point!

Abstract indeed! Here are two relevant quotes:

Mathematics, rightly viewed, possesses not only truth, but supreme beauty — a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show.

- Bertand Russell

and

In building a statue, a sculptor doesn't keep adding clay to his subject. Actually, he keeps chiseling away at the nonessentials until the truth of his creation is revealed without obstruction.

- Bruce Lee

As for applying intuition, realize that you can create new topological spaces using the direct sum. So a space can be built up as the disjoint (possibly infinite) union of completely separate and disparate parts.

Also, I found two math.stackexchange posts that might be helpful:

How to prove there exist a homeomorphism between any open ball and the entire normed space

So every n-dim Euclidean space is homeomorphic to any open ball, and that is easy to picture.

Now think about a topological space with a generating base of homeomorphic 'chunks of space'. If you take n-dim Euclidean space and start removing points, lines, lower dimensional 'planes' you may still be able to find open n-dim balls of space around all the remaining points. But what happens if you start removing open 'n-dim' balls? Is it still generated by open Euclidean n-balls?

Find a topological space that is not homeomorphic to the disjoint union of its connected components

So you can't even say that every topological space is the direct sum of connected spaces. But at least when there are only a finite number of connected components you can intuitively regard the space as an 'orthogonal sum' of those pieces.

Answer 2

General Topology is loosely a generalization of geometry, so I would say that the best way to get a grip on (General) Topology, and learn to get to get intuition for General/Point-Set Topology, is to learn how to visualize what is going on.

And the best way to visualize what's really going on, is to draw pictures of the definitions and concepts that frequently pop up.

For the last few concepts of neighbourhoods, interior point, interior, boundary, boundary point, these are all really visually intuitive if you draw pictures of them out. As an exercise try to come up with drawings to encapsulate these definitions.

For example a normal space is a topological space, where for every two disjoint closed sets, $E$ and $F$, there exists disjoint open $U$ and $V$ containing $E$ and $F$ respectively.

This can be easily encapsulated by the following drawing (courtesy of Wikipedia)

Finally I should add, that drawing pictures is merely a means to gain intuition, but it is something that you should not shy away from.