What is point-set topology?
Point-set topology is the most basic kind of topology. The most important ideas in topology are continuity, compactness, and connectedness. The goal of topology is to understand these properties for a wide variety of spaces. The intuitive definitions of these three things are:
- Continuity: A function is continuous if it takes nearby points to nearby points.
- Compactness. A space is compact if it is the union of a finite number of arbitrarily small sets (we usually say it is ‘covered’ by these sets).
- Connectedness. A space is connected if it cannot be split up into two pieces that are distant from each other.
Each of these concepts had weasel words: nearby, small, and distant. What do these things mean?
It turns out that they can all be made precise by using one idea: open sets. If two points are in a lot of the same open sets, they are close. If they are not in a lot of open sets, they are far away.
So we have more precise definitions:
- Continuity: a function f is continuous if the preimage of every open set is open.
What does this mean? Let U be an open set around any image point f(x) (so U can be as small as we want). Then f is continuous if for every such U, there is an open set V in the domain that contains x and that maps into U. This is the epsilon-delta definition without numbers.
- Compactness: A space X is compact if given any collection of open sets that cover X (no matter how small each one is), we can choose out a finite number of them that still cover X. (Notice that the real numbers aren’t compact because if you choose really small open sets, like the open sets (n,n+1), you need infinitely many to cover the whole line. The circle, though, is compact).
- Connectedness: A space is connected if you cannot split it into two disjoint open subsets (because each one being open means that they are far from each other).
So the question remains: What is an open set? And here is the secret of topology: an open set can be anything you want. So you just pick the open sets, and see what properties come out. Each choice of open sets is called a topology.
Almost anything goes when picking topologies, but for basic concepts to make sense (like constant functions being continuous) you need 3 basic rules; 1. The whole space is open, and the empty set is open. 2. The intersection of two (or any finite number) of open sets is open. 3. The union of any collection of open sets is open.
Pick weird topologies, and you get weird properties. Like the open ray topology; if the open sets are of the form (-infinity,a) then every subset is connected. Weird! Usually, though, you pick topologies that resemble Rn or other well known spaces. For fun reading, check out the book Counterexamples in Topology.
The image at the top of this post is one I created for the compact set page on Wikipedia.
This post was originally posted on reddit.