Subspace topology

For $A\subseteq X$, the subspace topology consists of the intersections $A\cap U$ with open subsets $U\subseteq X$.

A subset $B$ of a topological space $X$ is locally closed if $B=U\cap F$ for an open set $U$ and a closed set $F$. Equivalently, $B$ is open in its closure $\overline{B}$, or $\overline{B}\setminus B$ is closed in $X$. For the equivalence, $B\subseteq F$ implies $\overline{B}\subseteq F$, so $B=U\cap \overline{B}$. Relative complementation proves the second condition implies the third. Conversely, if $C=\overline{B}\setminus B$ is closed, then $B=(X\setminus C)\cap \overline{B}$ is such an intersection.

A topological subspace of a topological space $X$ is a subset $A\subseteq X$ equipped with the subspace topology inherited from $X$.

A closed subspace of a topological space $X$ is a closed subset of $X$ equipped with its subspace topology.