The sets $X = \lbrace (x,y) \in \mathbb{R}^{2}| y = 0, 0<x<1 \rbrace$ and $Y = \lbrace (x,y) \in \mathbb{R}^{2}|y=0 \rbrace$ are homeomorphic but there are no homeomorphism $h: \mathbb{R}^{2} \longrightarrow \mathbb{R}^{2}$ such that $h(X) = Y$
I don't know how to start this question. I don't want a solution, just a idea, a hint.
1) If you can get an homeomorphism $f\colon ]0,1[ \to \Bbb R$, then $(x,0) \mapsto (f(x),0)$ is an homeomorphism between $X$ and $Y$.
2) If $h\colon \Bbb R^2 \to \Bbb R^2$ is an homeomorphism, then $h|_{\Bbb R^2 \setminus X}: \Bbb R^2 \setminus X \to \Bbb R^2 \setminus Y$ will also be one. Argue that this cannot happen (for example, using connectedness).