I think of topological spaces as coming in several "islands of interestingness" (the CW island, the Zariski archipelago,...) dotting a vast "pathological sea" (the long line ocean, the gulf of the lower limit...). That is, I only know how to think about a topological space if it happens to live on one of these islands, the methods appropriate to one island may be completely unrelated to those of another, and a "random" topological space is probably unrelated to anything I know how to think about and is thus "pathological". I'd like to get a better perspective on how many of these islands there are -- and perhaps whether some which I think are distinct are actually connected by some isthmus. Here's what my current map looks like:
CW complexes (and spaces homotopy equivalent to such)
Zariski spectra of commutative rings (and schemes)
Stone spaces (totally disconnected compact Hausdorff spaces)
Infinite-dimensional topological vector spaces (and spaces locally modeled on them)
One of the characteristic features of this map is that
there is little overlap between the islands although there is overlap between these islands in a literal sense, what really sets them apart is that the tools used in exploring one island bear little resemblance to those used for another. For example, when studying spaces using CW complex tools, non-Hausdorffness is regarded as pathological, clopen sets as uninteresting, and infinite-dimensionality as an annoyance whereas such features are respectively embraced when studying Zariski spectra, Stone spaces, and functional-analytic spaces.
Are there other classes of topological spaces which are interesting to study (and not just as a source of pathologies)?
Are these islands less isolated than I'm making them out to be? E.g. are there interesting topological considerations to be made which apply simultaneously to, say, Banach spaces and Stone spaces?
Is it correct to think that the ocean is vast, i.e. that "most" topological spaces are "pathological"?