When is the union of two locally closed subsets locally closed?

Question

It isn't true in general that two locally closed subsets (i.e., subsets of the form open $\cap$ closed) are locally closed, so is there some standard condition that guarantees it?

Answer 1

Yes, by the following:

Let $A, B$ be locally closed subsets of a topological space $X$. Then: $\overline{A} \cap \overline{B} = \emptyset \Rightarrow \overline{A} \cap B = A \cap \overline{B} = \emptyset \Rightarrow A \cup B$ is locally closed.

Proof: The first implication is clear. For the second one, let $U, V$ be open sets such that $A = U \cap\overline{A}$ and $B = V \cap\overline{B}$. Then $A \cup B = ((U \setminus \overline{B}) \cup (V \setminus \overline{A})) \cap \overline{A \cup B}$.