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

Question

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.

Thanks in advance.

Answer 1

The fundamental group does not, in fact, distinguish spaces up to homeomorphism.

For a simple example of this, each of the following spaces have trivial fundamental group, yet no two are homeomorphic:

• The real line, $\mathbb{R}$.

• The "Long line" https://en.wikipedia.org/wiki/Long_line_(topology).

• The plane $\mathbb{R}^2$.

• The one-point space.

• The 2-sphere $\{(x, y, z)\in\mathbb{R}^3: x^2+y^2+z^2=1\}$.