I know that topological spaces over a given finite set are classified. But what about sets with cardinality $\omega$? The only restriction I would require is for the spaces to be $T_0$, i.e. have no topologically indistinguishable points.
I initially thought that it would be sufficient to look at connected spaces, but then I realized that just because a space is disconnected, it need not be completely decomposable into connected components: Consider, for instance, $\{1/n\mid n\in \mathbb N_{>0}\}\cup \{0\}\subseteq \mathbb R$ equipped with the subspace topology: Every set of the form $[0,1/n]$ is clopen, but as their intersection $\{0\}$ is not open, there is no minimal clopen set around $0$, i.e. no connected subspace $S\subseteq X$ that admits a decomposition $X \simeq S \sqcup X\setminus S$.
There is a classification for countable metric spaces (I sketch it in my answer to this related question, they include all the countable ordinals and the rationals and combos of those), but beyond that it's isolated results: there are the 4 countable Toronto spaces that are not indiscrete, included and excluded point topologies, many different types of spaces based on an filters on $\omega$ (we already have more than continuum many homeomorphism types there), either using the filter as the topology, or its elements for a single non-isolated point aded to $\omega$ (but there could be more of those of Hausdorff is not required) etc etc. I also don't know what the homeomorphism type of a dense countable set of $\{0,1\}^\mathbb{R}$ is, and how many types there are (they are all crowded normal and of weight continuum). Etc. etc. Arens space is countable and quite interesting. The one-point compactification of the rationals, and countable connected Hausdorff spaces also exist. In $\pi$-base you can look at more examples (often introduced to show specific combinations of properties are possible). There is much more variation than you'd think at first. A full classification seems far off as yet.