Locally closed subset

In topology, a branch of mathematics, a subset ${\displaystyle E}$ of a topological space ${\displaystyle X}$ is said to be locally closed if any of the following equivalent conditions are satisfied:^[1][2][3][4]

• ${\displaystyle E}$ is the intersection of an open set and a closed set in ${\displaystyle X.}$
• For each point ${\displaystyle x\in E,}$ there is a neighborhood ${\displaystyle U}$ of ${\displaystyle x}$ such that ${\displaystyle E\cap U}$ is closed in ${\displaystyle U.}$
• ${\displaystyle E}$ is open in its closure ${\displaystyle {\overline {E}}.}$
• The set ${\displaystyle {\overline {E}}\setminus E}$ is closed in ${\displaystyle X.}$
• ${\displaystyle E}$ is the difference of two closed sets in ${\displaystyle X.}$
• ${\displaystyle E}$ is the difference of two open sets in ${\displaystyle X.}$

The second condition justifies the terminology locally closed and is Bourbaki's definition of locally closed.^[1] To see the second condition implies the third, use the facts that for subsets ${\displaystyle A\subseteq B,}$ ${\displaystyle A}$ is closed in ${\displaystyle B}$ if and only if ${\displaystyle A={\overline {A}}\cap B}$ and that for a subset ${\displaystyle E}$ and an open subset ${\displaystyle U,}$ ${\displaystyle {\overline {E}}\cap U={\overline {E\cap U}}\cap U.}$