Definition - A subset $Y$ of a topological space $X$ is said to be locally closed if it is the intersection of an open and a closed subset.

The following result provides some equivalent definitions:

Proposition - The following are equivalent:

1. $Y$ is locally closed in $X$ .
2. Each point in $Y$ has an open neighborhood $U \subseteq X$ such that $U \cap Y$ is closed in $U$ (with the subspace topology).
3. $Y$ is open in its closure $\overline{Y}$ (with the subspace topology).