homeomorphic to [-1,1]

Question

It's written that one point compactification of $[−1,1]\setminus\{0\}$ is homeomorphic to $[-1,1]$ in my topology book. I want to know why they are homeomorphic.

Answer 1

Fact:

If $X$ is locally compact Hausdorff and we have a compact Hausdorff space $Y$ such that $X \subseteq Y$ and $Y\setminus X$ consists of a single point $p$, then $Y$ is homeomorphic to the one-point compactification $\alpha X$ of $X$.

Apply this to $X=[-1,1]\setminus\{0\}$ and $Y=[-1,1]$.

To prove the fact, let $\alpha X = X \cup \{\infty\}$ in its usual topology. Check that $f: Y \to \alpha X$ defined by $f(x)=x$ for $x \in X$ and $f(\infty) = p$ is a continuous bijection and then compactness (and Hausdorffness) makes this a homeomorphism.

A slight generalisation of the fact: if $Y$ is compact Hausdorff and for some point $p \in Y$ we have that $Y\setminus\{p\}$ is homeomorphic to $X$, then $Y$ is homeomorphic to the one-point compactifcation of $X$. I.e. if $Y$ is compact and one point larger than something homeomorphic to $X$, it must be its one-point compactification. It shows this compactification is essentially unique.

Answer 2

No, they are not homeomorphic, the second set is connected, but not the first.