Locally compact hausdorff subspace is open in compact Hausdorff space??

Question

In the book ‘Topology’ by munkres, there’s one theorem.

If $X$ is locally compact hausdorff space if and only if there exists a space $Y$ satisfying followings:

1. $X$ is a subspace of $Y$
2. $Y\setminus X$ is a single point
3. $Y$ is compact hausdorff space.

And $Y$ is unique up to homeomorphism.

In the part of uniqueness proof, the author shows if there exist two such spaces, say $Y$ and $Y'$, then there’s homeomorphism between them. In the process, he uses the fact that $X$ is open in $Y$. I don’t understand the reason why it’s open in $Y$.

I think subspace needs not be open in the original space.

Thank you for your help in advance

Answer 1

$Y \setminus X$ is a single point, but singletons are closed in a Hausdorff space (and $Y$ is Hausdorff). So, the complement of $X$ is closed in $Y$ hence $X$ is open in $Y$.