Is this proof for Corollary 29.4. Munkres Topology correct?

Question

The followings are from Munkres' Topology:

After a long time I still can't prove Corollary 29.4. based on the information from Corollary 29.3. and Theorem 29.1. especially when they are irrelevant (note that Y-X must be a single point).

Please help!

Answer 1

First, suppose $X$ is homeomorphic to an open subspace $A$ of a compact Hausdorff space $Y$. By Corollary 29.3, $A$ is locally compact, and hence so is $X$. Also, $A$ is Hausdorff since any subspace of a Hausdorff space is Hausdorff, and thus so is $X$.

Conversely, suppose $X$ is locally compact Hausdorff. By Theorem 29.1, there exists a compact Hausdorff space $Y$ such that $X$ is a subspace of $Y$ and $Y-X$ has one point. Since $Y$ is Hausdorff, the singleton set $Y-X$ is closed in $Y$, so $X$ is open as a subset of $Y$. Thus $X$ is homeomorphic to an open subspace of a compact Hausdorff space (namely, itself as a subspace of $Y$).

Answer 2

The space $Y$ in theorem 29.1 should be the one-point compactification (Alexandroff extension) $X^* = X \cup \{\infty\}$ of $X$. The inclusion $X \rightarrow X^*$ is always continuous and open; $X^*$ is always compact; and it is Hausdorff if and only if $X$ is locally compact and Hausdorff.