Homeomorphism on $\mathbb{R}^{2}$. Hint.

Question

Prove that the sets $X = \lbrace (x,y) \in \mathbb{R}^{2}\,|\,y=0\;\text{and}\;0<x<1 \rbrace$ and $Y = \lbrace (x,y) \in \mathbb{R}^{2}\,|\,y=0 \rbrace$ are homeomorphics but there is no homeomorphism $h: \mathbb{R}^{2} \to \mathbb{R}^{2}$ such that $h(X) = Y$.

The first afirmation is easy, but I don't know how to show the second afirmation. Any hint?

Answer 1

Consider $Z=\bigl\{(x,y)\in\Bbb R^2\mid y=0\text{ and }0\le x\le 1\bigr\}.$ How is $Z$ related to $X$? How is $h(Z)$ related to $h(X),$ then, if $h$ is a homeomorphism $\Bbb R^2\to\Bbb R^2$? If $h$ were a homemorphism $\Bbb R^2\to\Bbb R^2$ such that $h(X)=Y,$ then what would $h(Z)$ be? What can we then conclude?

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Added: Put in a much more straightforward fashion, note that $Y$ is closed in $\Bbb R^2.$ Can we say the same about $X$?

Answer 2

Here's a hint: Is either $X$ or $Y$ open/closed in $\mathbb R^2$?