Are open or closed sets locally closed?

Question

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.

Answer 1

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: $$\textit{ If $Y$ is a locally compact space and $X\subset Y$ is locally closed, then $X$ is locally compact as well.}$$