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I have two equivalent definitions of a locally closed subspace $X$:

  1. $X$ is closed in its closure $\overline{X}$ with respect to the subspace toplogy.
  2. $X$ is the intersection of an open and a closed set.

If $X$ is open, then $X\cap\overline{X} = X$, thus $X$ is the intersection of an open and a closed set, thus locally closed. If $X$ is closed, then $\overline{X} = X$, thus $X$ is closed in its closure.

My answer therefore tends to be yes, but I just want to make sure.

Thank you in advance.

Marc
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  • Note that the first statement says $X$ closed in $\overline{X}$, not necessarily closed in the ambient topological space. So you cannot assume $X=\overline{X}$ – Frank Lu Feb 23 '15 at 16:42
  • Sorry, I am afraid I don't understand. Suppose we have a topological space $A$ and $X\subseteq A$ is closed. Then we do have $X=\overline{X}$ right? – Marc Feb 23 '15 at 16:47
  • Wait. If $X$ is closed in its closure, then it is closed in the ambient space. You probably mean "open in its closure" ? – Stefan Hamcke Feb 23 '15 at 16:49
  • AH! Thank you, my bad, you are completely right. The definition seemed so useless liket this^^ – Marc Feb 23 '15 at 16:54

1 Answers1

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You are right. Note that you can always write $X=X\cap Y$, where $Y$ is the underlying space, which is always open and closed in $Y$. So if $X$ is open (resp. closed), then it is the intersection of an open (resp. closed) set with a closed (resp. open) set.

The first definition should certainly read: X is open in its closure, since closed in its closure would imply that $X$ is closed.

By the way, here's an application of such sets: $$\textit{ If $Y$ is a locally compact space and $X\subset Y$ is locally closed, then $X$ is locally compact as well.}$$

Stefan Hamcke
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  • Oh right, so simple. Thank you for your response. – Marc Feb 23 '15 at 16:58
  • @Stephan Hamcke Why if $X$ is closed in it's closure it means that $X$ is closed? – roi_saumon Jun 11 '20 at 17:22
  • Oh, I see. Because if $X$ is closed in $\overline{X}$, this means that $X = \overline{X}\cap C$ where $C$ is some closed set in the topological space. So then $X$ is a closed set since it is the intersection of two closed spaces. – roi_saumon Jun 11 '20 at 17:34