What does a topology tell you about the relationship between its elements?

Question

I'm new to topology, and I'm having some trouble. I understand that a donut and a mug have the same topology. In my intuition, that is because one of them can be stretched into the other, and all the points will still be connected to each other the same way as they were before(don't know if "connected" is the right term). What I don't seem to get my head around, is how a topology is giving you this info, about how the elements in a set are related/connected (if you can use that term). If I have the set $X = \{a, b, c, d\}$ with the topology $t = \{\{\}, \{a, b\}, \{a, b, c\}, \{a, b, c, d\}\}$, how do you interpret $(X, t)$ to know which elements are related/connected? I can't seem to find any information about how you really interpret a topology. Like what does a topology really tell you about the relationship of the elements in its sets? Is this a valid question, or am I missing something fundamental?

Answer 1

I meant for this to be a comment, but it got a little bit long so I'm leaving it here. If anyone wants to add a more comprehensive answer, or expand on this one, they should feel free to.

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Topology tells you what elements are "close to" each other, while avoiding a notion of "distance". It does this by providing a notion of open set, where if $x \in U \subseteq V$, we can think of elements in $U$ being "closer to $x$" than elements of $V \setminus U$. The intuition you want to have here is related to the open balls of a metric space. If we forget about the actual distance on $\mathbb{R}$, we can still remember that points in $(-1,1)$ are closer to $0$ than points in $(-2,2)$.

Closed sets too, provide "closeness" information. If $x \in \overline{A}$, that is, the closure of $A$, then $x$ is "almost in $A$". The picture to have in mind here is $A = \{ (x,y) ~|~ x^2 + y^2 < 1 \}$. The point $(2,2)$ is somehow "less in $A$" than the point $(1,0)$, even though both aren't in $A$. With naked sets, we cannot make this intuition precise, but with a topology, we can. The point $(1,0)$ may not be in $A$, but it is in $\overline{A}$. No matter how close to $(1,0)$ we want to get (as measured by open sets), we can always find a point that is close enough and also in $A$.

As an aside, we can dualize this argument. If we work with $B = \{(x,y) ~|~ x^2 + y^2 \leq 1\}$, then the point $(1,0)$ is somehow "less in $B$" than the point $(0,0)$. This is topologically expressed by $(0,0)$ being in the interior of $B$. We know $(0,0)$ is REALLY in $B$ because we can find a set of points close to $(0,0)$ (again, measured by an open set) where every point which is close enough is also in $B$.

As a test of understanding, you should take the standard definitions of interior and closure and show they align with the intuition I've given above.

"But", I hear you asking, "what about rubber-sheet geometry?" A circle and a square are homeomorphic (i.e., topologically the same) because you can find a bijection which respects "closeness" in each direction. By stretching the square into a circle, points which start close together stay close together, though the concrete distance between points changes. However, a circle and a filled in circle (i.e. a disk) are not homeomorphic because either you must tear the disk (and so points that started close together end far apart) or you must glue the circle to itself (so that the inverse function involves tearing).

Finally, to address the four point space you brought up, I have this (perhaps controversial) opinion: Finite topological spaces are interesting in their own right, but they aren't very good for building intuition. They CERTAINLY aren't good for working with intuition you may have about rubber-sheet geometry. It is somewhat unfortunate that a first course in topology is typically obsessed with pathologies and problematic spaces. There is good reason to do this! Knowing about pathologies is useful for understanding what can go wrong, and tempering the intuition that working with metrizable spaces for too long will give you. Topology is also incredibly useful outside of geometry, and the material of a point set topology class is extremely relevant to analysis.

This focus, however, obscures the connection between geometry and topology that many of us were promised by pop-math articles. Once you take a course in Algebraic Topology, you will see the connections you are hoping for (provided you have a good professor, of course).

If you want a jump-start on this material, I cannot recommend the book "A combinatorial introduction to topology" by Michael Henle highly enough. It is accessible without much background, and explains algebraic topology in a very polite way, while still managing to hit some fairly advanced material.

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I hope this helps ^_^