﻿I’m going to be talking about the main invariants of topological spaces. This is my topic list:


Separations properties: T1, T2 (Hausdorff), T2 an a half; covers and open covers, subcovers and finite subcovers; compact spaces; Heine-Borel theorem (compact if and only if closed and bounded); open and locally finite refinements; paracompactness; metrisable spaces and Stone's theorem; long line (or Alexandroff line); partition of unity subordinate to an open cover; examples; connectedness and proof that M is connected if and only if M and the empty set are the only subsets which are both open and closed; path-connectedness and proof that path-connectedness implies connectedness; homotopic curves on a topological space; concatenation of curves; fundamental group; group isomorphism; topological invariants and classification of topological spaces; examples: 2-sphere, cylinder, 2-torus.


The TLDR is that topological spaces are such a large category that they’re unclassified. If you give me a topological space (X, O_x) and (Y, O_y), and start listing off invariants that both of them have in common, there is no definitive list you can make that guarantees that X and Y are homeomorphic: ie that there exits a homeomorphism: ie that there exists an invertible, in both ways continuous map: ie that you can mold you topological space visually into the other (note this visualization works only if you’re using the standard topology in a manifold that can be embedded in R^3.


Some invariants measure some type of coarseness or fineness of the topology at different locality levels. The first one does exactly this.


One of the first invariants we encounter are the separation properties. Have T0, T1, T2 etc until higher T’s. These basically measure the fineness of your topology. They can all be formulated in terms of having two points, and how many open sets and closed sets you have around each point without these open/closed sets being forced to include both points.


T1 says you have at least one open set around one of the point that doesn’t include the second point.


T2 says you can have open sets around both points such that the open sets don’t intersect.


Almost every single intuitive topological spaces you think of are h2 or higher.


Our next invariant is compactness and paracompactness. Again these are properties almost all of our intuitive examples of topological spaces contain. Compactness measure the closeness globally of a space, so R^n is not compact, but S^n is.


Compactness is measured in terms of every cover having a finite subcover. A cover is a collection of open sets. A subcover is the subset of the cover. Compactness is then a very strong condition, because if you find even a single counterexample, then the topological space fails to be compact.


The classic example is with [0,1] not being compact, but (0,1) being compact. The tldr is that you can create an infinite series that when unioned together in a cover, will cover the [0,1], but that it becomes impossible to create a finite subcover.


The heiner-borel theorem says that any subspace R^n is compact only if it is closed and bounded. The result also generalizes for metric spaces, since metric spaces induce topological spaces.


Paracompactness is similar, but less powerful, since even R^n is paracompact. It basically enforces compactness locally.


Stone's theorem says every metrizable topological space is paracompact.


You then have partitions of unity, which are going to end up being quite useful when we decide to integrate on manifolds and want to determine how much each chart contributes to a viewpoint when two charts intersect.


We then have connectedness, which is quite obvious and just means the space is volume connected.


Path connectedness is the same, but that it’s connected by path on its surface.


Both of these are invariants.


Finally you have the fundamental group, which is the most interesting of all the invariants. It asks questions about loops on the surface of the topological space, and what the “symmetries” of these loops are. In other words what group they form. Apparently this group is an invariant.


Setting up the mathematical framework for talking effectively about loops can be quite cumbersome though.