# Does fundamental group distinguish between any two non homeomorphic topological space?

I am new to fundamental group.

I was reading Munkres and found that need of fundamental group was to distinguish between non-homeomorphic topological spaces.

So my question is, does fundamental group distinguish between any two non-homeomorphic topological space?

Or there exist some spaces which are non-homeomorphic but their fundamental groups are same?

My intution says it's a successful tool to distinguish between them.

• Any two homeomorphic topological spaces have the same fundamental group (and indeed the same homotopy groups), but homotopy groups alone (let alone just the fundamental group) cannot disinguish any two nonhomeomorphic spaces. – florence Jul 3 '16 at 12:40

• @Human There are many other invariants. For instance, the higher homotopy group $\pi_2$ can distinguish the $2$-sphere from the others. Then there are invariant properties like compactness, connectedness, etc. More complicated arguments are also possible: the real line becomes disconnected when any single point is removed, which distinguishes it from the plane; and the long line does not have a countable dense subset, which distinguishes it from the others. But topological spaces are "wild": there isn't a simple invariant (or set of invariants) which characterizes them up to homeomorphism. – Noah Schweber Jul 3 '16 at 12:42