this is a relatively broad question I will try to formulate it as clearly as possible. If we are given a topology $\tau$ on a set $E$ and an algebraic structure on this set, we know we can make this structure "topological" in some way if we allow the continuity of the algebraic operations.
My question is: Is it possible to obtain algebraic structures by purely topological constructions? In other words, are there some topologies on a space that can allow us to recover some algebraic structures on that space?
EDIT: The question could be reformulated more precisely as: are there natural "algebraic" categories, with a faithful functor to the category of topological spaces which reflects isomorphisms