In my attempt to understand the first paragraph and the notion of a topology (and a topological space) I realized that there was something I wasn’t getting. I understood the definition of “open” subsets of X, I also understood the meaning of “neighborhoods,” and I felt like I was getting a grasp on the axioms.
But what does it mean to have a “notion of nearness”?
Now, I think I’m starting to get it. In Calculus or Real Analysis you define a continuous function in terms of distance and the “I can get closer” game. But the point is that you don’t need distance in order to define continuity or a lot of those other analytic notions. All you need is a “notion of nearness,” and if open sets and neighborhoods make sense in your space, then you can usefully obtain that notion of nearness.
The usual Calculus definition of a continuous mapping goes like this: if we want all the values in the range to be within a certain neighborhood — no matter how narrow the neighborhood — we can always find a suitable neighborhood in the domain whose points all map to that band of points in the range. In this, a neighborhood is defined in terms of distance — wanting to be in a certain neighborhood in the range is being a certain (potentially tiny, epsilon) distance away from a point in the range.
But who needs to define a neighborhood in this way? We can replace this metric notion of neighborhood with a topological one, the one that we struggled to grasp in the previous post, that really comes down to open subsets. So, in short, a function is continuous at a point in a topological space if, no matter the neighborhood in the range (call it Y), there’s always a neighborhood in the domain (X) that the function maps to a neighborhood that’s a subset of the original (Y).
I should probably be trying to make these images myself rather than stealing them from wikipedia but whatever:
So now I’m thinking of topological spaces as metric spaces, minus distance. Which explains something I never really understood, which is why in topology you can deform all sorts of things into other things. Why should a coffee mug and a doughnut be equivalent in a topological sense? The answer is that distance doesn’t exist in a topological space. All we have is open sets and subsets of open sets which stands in for our notions of nearness. As long as a mapping (like a homeomorphism) preserves the open set structure — the idea that things that start out near each other should remain near each other, relatively speaking — then there is a topological sameness that is preserved.
Honestly I should probably just be learning topology instead of trying to make sense of this algebraic topology text, but let’s see how far we can go. The next paragraph makes the jump to algebraic topology.