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I recently read something along the lines of "General topology deals with the nice properties of pathological spaces, and algebraic topology with the pathological properties of nice spaces".

In a sense, this is consistent with my experience of both subjects, at least the part concerning which spaces we're talking about : in general topology often one thinks of the worst possible cases (e.g. totally disconnected spaces, non separated spaces, etc.), but in algebraic topology those are usually kept out : if someone says "but the space has to be connected", in algebraic topology we'll gladly add this as a hypothesis without a second thought; often we deal only with CW-complexes, or "connected, locally path-connected, semi-locally simply connected spaces" which are very far from the beasts one is concerned about in general topology.

Of course this is very schematic and I'm just generalizing stuff (though if you think this is a bad generalization, please tell me so !), especially about general topology.

However it's easy to come up with natural statements concerning nice spaces in general topology, we can easily add hypotheses such as path-connectedness etc.

But can we go in the other direction ? Is there some "algebraic topology of pathological spaces" ?

Of course it would have to look very different from what algebraic topology usually is (at least the one I know) : for instance the fundamental group of a totally disconnected space is quite useless.

But I can't see a reason why, a priori, one couldn't associate interesting algebraic invariants to pathological spaces, perhaps they wouldn't be groups, but some other thing. For a more formal question (though still vague) : is there a reason why there aren't nice functors from $\mathbf{Top}$ to "algebraic" categories ? (I'm adding the "category theory" tag because there might be an answer concerning the structure of $\mathbf{Top}$ that would hinder the existence of such functors)