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Is there an established name for a subset of the form $U \cap V$ where $U \subset X$ is open and $V \subset X$ is closed? For example, locally compact subspaces of a locally compact Hausdorff space are exactly of this kind. If there are no existing names, I welcome suggestions for good names (in comments, because that does not fit the question-answer format).

kaba
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    Such sets are usually called locally closed, i.e. "$U\cap V$ is a locally closed subset of X". – user520682 May 17 '20 at 14:24
  • Nice! Googling this term I found out that locally closed is equivalent to that for each $x \in U \cap V$ there is an open neighborhood $U_x$ of $x$ such that $U \cap V \cap U_x$ is closed in $U_x$. Hence, this is a good name. I haven't made this connection before. Perhaps make it an answer? – kaba May 17 '20 at 14:47

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As user Joo correctly said, the usual term for a set $Y$ which is intersection of an open set and a closed set is "locally closed". You've realised that it's meant to be understood as the equivalent statement:

For all $x\in Y$ there is an open set $U\subseteq X$ such that $x\in U$ and $Y\cap U$ is closed in $U$.

You should be aware of a little trap. Recall that there is a lemma of general topology establishing that open covers are fundamental. In other words: for a topological space $X$, a subset $Y\subseteq X$ and a family $\mathcal F$ of open subsets of $X$ such that $\bigcup \mathcal F=X$, we have that $Y$ is closed if and only if $Y\cap U$ is closed in $U$ for all $U\in \mathcal F$.

This result implies the following:

A subset $Y$ in a topological space $X$ is closed if and only if for all $x\in X$ there is an open set $U\subseteq X$ such that $x\in U$ and $U\cap Y$ is closed in $U$.

So be aware of the difference and don't be tricked! $\ddot\smile$

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    That's a very good warning, and a nice association:) It took me many reads to spot the change from $Y$ to $X$, although I knew what you were getting at. – kaba May 17 '20 at 15:11