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.
perhaps he is implying some even stronger result
He is referring to the following result of Peter Freyd (Freyd uncertainty principle):
The homotopy category of spaces $HoTop$ does not admit a faithful functor to the category of sets $Set$. Specifically, for any functor $T: Top_* \to Set$ from base-pointed spaces to sets which is homotopy invariant, there exist a triple $f: X \to Y$ such that $f$ is not null-homotopic, but $T(f) = T(\ast)$. Here $\ast$ is the null map to the basepoint of $Y$.
In particular, any algebraic invariant is a set-valued homotopy invariant. This includes homotopy groups, cohomology, cohomology and homotopy operations and whatever you can think of. Freyd's theorem implies that we cannot describe the homotopy category as a category of algebras for some algebraic theory $\mathcal{T}$, since any algebraic category is concrete.
Fun fact: Freyd's theorem essentially relies only on the general set-theoretic arguments and cardinal counting.
Non-counter-example: Whitehead's theorem states that if a map $f: X \to Y$ between pointed connected CW-complexes induces an isomorphism on all homotopy groups $\pi_i, \ i=1, 2,\dots$, then $f$ is a homotopy equivalence between $X$ and $Y$. Note that you still can't discriminate spaces looking just at the collection of homotopy groups: there can be $X$ and $Y$ such that $\pi_i(X) = \pi_i(Y)$ for all $i$, but this isomorphism is not induced by any actual map $F: X\to Y$ and the spaces are not homotopy equivalent. The simplest example is $\Bbb R \Bbb P^2 \times S^3$ and $S^2 \times \Bbb R \Bbb P ^3$. They both have a double covering by $S^2 \times S^3$ and have thus the same homotopy groups, but their cohomology is not isomorphic.
On second thought, I can't see why Freyd's theorem would imply actual non-discriminable spaces without any extra conditions on the invariant. Perhaps someone can fill this gap, but imho the non-discrimination of maps is bad enough.
Since this theorem states indiscriminability even for non-homotopy equivalent spaces, it in particular does so for non-homeomorphic spaces. This could be weaker than what your professor implied since we could in principle consider invariants of spaces which are not homotopy invariants. However, this requires some more specifics on what we would call "algebraic invariants" since e.g. the lattice of open subsets looks like a perfectly fine algebraic invariant to me, but it certainly discriminates spaces.
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.
Topological spaces can be infinitely varied. From topology point of view even something as deadpan as $\mathbb{R}$ has an enormous amount of structure.
What about the Zariski topology in algebraic geometry? They tend to be different than the topology you inherit from their embedding into ambient space.
