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


1 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 $Y$.

  • $\begingroup$ Thank you so much, but could you explain why singletons are closed in a Hausdorff space? $\endgroup$
    – ylh0501
    Nov 27, 2017 at 4:36
  • 1
    $\begingroup$ I'll refer you to math.stackexchange.com/questions/2495812/…. It's a standard fact about Hausdorff spaces, and worth knowing! $\endgroup$
    – B. Mehta
    Nov 27, 2017 at 4:37

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