TITLE: Algebraic topology, proving two spaces aren't homeomorphic
QUESTION [5 upvotes]: I need help on how to approach a problem of this kind, I'm given two topological spaces:
$$X=\mathbb{R}^2-\{(n,0)|n\in\mathbb{N}\}\text{ and }Y=\mathbb{R}^2-\{(\frac{1}{n},0)|n\in\mathbb{N}\}$$
I want to show that they're not the same but I have no clue where to start, one way could be showing that $\pi_1(X)\neq\pi_1(Y)$, but I don't know how to compute those groups.

REPLY [4 votes]: In $Y$, there exists a point such that no neighborhood of the point is simply connected. Since $X$ does not have this property, they cannot be homeomorphic.