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:
- $X$ is a subspace of $Y$
- $Y\setminus X$ is a single point
- $Y$ 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