TITLE: No matter how many algebraic invariants we attach to topological spaces, there will always be nonhomeomorphic spaces agreeing on all their invariants
QUESTION [33 upvotes]: A while ago a professor of mine said something along the lines of

No matter how many algebraic invariants we attach to topological spaces, there will always be nonhomeomorphic spaces agreeing on all their invariants

Where algebraic invariants are functors from $Top$ to $Grp$ I guess. I've not been able to find this result anywhere online (partly because I don't know what to google), so I'm looking for a source where this result is discussed.

REPLY [3 votes]: There are interesting jumps in the responses from the question to other questions, e.g. from homeomorphisms to homotopy equivalences, from spaces to pointed spaces, .... One reason for the homotopy  study is that invariants up to homeomorphy might be uncountable, except for restricted classes of spaces, while for classes of "nice" spaces we can hope for a countable number of homotopy invariants. 
In my own work I have been forced into considering "structured spaces" to get higher homotopy groupoids,  and so colimit theorems, in higher homotopy; this pdf file on "A philosophy of modelling and computing homotopy types" is a relevant 2015 presentation. These notions of structured spaces, i.e. of topological data,  extend to higher dimensions the idea of using many base points for the fundamental groupoid, and do lead to some not previously available precise nonabelian colimit calculations of homotopy types, and so of homotopy groups, of pointed spaces. It should not be forgotten that homotopy groups are but a pale shadow of pointed homotopy types. 
Part of the philosophy is that in order to specify a space you need some kind of data, and that data will have some kind of structure. It could be wise to keep that structure in mind when constructing "invariants". It is for example standard to consider the cellular chain complex of a cell complex, and this is extended to crossed complexes and filtered spaces in the book Nonabelian Algebraic Topology. Grothendieck in Section 5 of his "Esquisse d'un Programme" argues that the notion of topological space is too far from the geometry, and wants more structure, such as a stratification.