Are locally compact Hausdorff spaces with the homeomorphic one-point compactification necessarily homeomorphic themselves?

Question

When practicing old qualifying exam problems, I had trouble with this one. Thanks for any help! Is it true that if the $1$-point compactifications of two locally compact Hausdorff spaces $X$, $Y$ are homeomorphic, then $X$ and $Y$ are necessarily homeomorphic? Give a proof or counterexample, as appropriate.

Answer 1

Always think about the simplest cases first! What possible one point comp. could give rise to, say, [0,1]?

Answer 2

Sharkos's answer tells the story simply and eloquently. I think it's worth noting that the converse does hold. That is, if $X$ and $Y$ are homeomorphic, then their respective one-point compactifications $\alpha X$ and $\alpha Y$ will be homeomorphic as well. Indeed, let $f:X\to Y$ be a homeomorphism and let $g:Y\hookrightarrow\alpha Y$ the canonical inclusion. Show that $g\circ f:X\to\alpha Y$ is continuous and injective. Since $X$ is dense (when canonically included) in $\alpha X,$ then there is a unique continuous function $h:\alpha X\to\alpha Y$ whose restriction to $X$ is $g\circ f$. Show that $h$ is a homeomorphism. You may need to consider separately the case where $X$ and $Y$ are compact Hausdorff and the case where they are not (one is trivial).