Say $X^{*}$ represents the one-point compactification of a space $X$ that is locally compact, non-compact, and Hausdorff. Also suppose that $Y$ is compact Hausdorff and $Y-\{p\}$, for a single point $p$, is homeomorphic to $X$. So then would $Y$ be homeomorphic to $X^{*}$? I think it is, but I'm not sure how it can be shown.
My attempt so far: Correct me if I'm wrong, but I started with inclusions $i: X \rightarrow X^{*}$ and $j: Y-\{p\} \rightarrow Y$, along with the homeomorphism $h: X \rightarrow Y-\{p\}$. So we need to find a homeomorphism $g: X^{*} \rightarrow Y$. So I tried using a commutative diagram and so far, I've concluded that $g$ (if it is to exist) would have to be continuous since the continuous image of a compact space is compact (I think this follows from $X^{*}$ being compact), but then one of my inclusions, i.e. $i$, can't possibly be continuous, can it (since $X$ is not compact)? How can this be fixed?