Can we rederive the axioms of topology from the structure of the category $\textbf{Top}$?

The approach of category theory to the description of mathematical structures, is to look at how a class of mathematical structures relate to each other, and to forget the structure itself.

e.g. in the category of topologies Top, objects are topological spaces, and morphisms are continuous functions. From a categorical perspective, Top only contains this information about the continuous functions, and forgets the "internal structure" of each object, i.e. the topological spaces.

I have read that it is an interesting property of category theory that this relational information "captures" the information about topological spaces.

Does this mean that we have literally all information about topological spaces in its category? e.g. can we rederive the axioms of topology from purely the categorical information in Top?

If not, then what does it mean concretely to say that the category $\textbf{Top}$ "captures what a topology is"?

Solution 1:

Let $P$ be the one point topological space, and let $U=\{u_1,u_2\}$ be the two-point topological space with open sets $\{\}$, $\{u_1\}$, and $\{u_1,u_2\}$. Then if $X$ is an arbitrary topological space, the set of points of $X$ is identified with $\mathrm{Hom}(P,X)$, and the set of open sets of $X$ is identified with $\mathrm{Hom}(X,U)$ (if $f:X\to U$ is continuous, then $f^{-1}(u_1)$ is an open subset of $X$, and every open subset comes from a unique $f$).

So we can "see" the internal structure of a topological space from just the categorical structure. With some care, every topological property can be phrased in terms of continuous maps. For example, a subspace of $X$ is an (isomorphism class of a) continuous map $f:Y\to X$ such that $\mathrm{Hom}(P,Y)\to \mathrm{Hom}(P,X)$ is injective and $\mathrm{Hom}(X,U)\to\mathrm{Hom}(Y,U)$ is surjective.