Locally closed subset

In topology, a branch of mathematics, a subset $E$ of a topological space $X$ is said to be locally closed if any of the following equivalent conditions are satisfied:

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

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

Examples

The interval $(0,1]=(0,2)\cap [0,1]$ is a locally closed subset of $ℝ.$ For another example, consider the relative interior $D$ of a closed disk in ${ℝ}^3.$ It is locally closed since it is an intersection of the closed disk and an open ball.

On the other hand, $\{(x,y)\in {ℝ}^2\mid x\ne 0\}\cup \{(0,0)\}$ is not a locally closed subset of ${ℝ}^2$.

Recall that, by definition, a submanifold $E$ of an $n$-manifold $M$ is a subset such that for each point $x$ in $E,$ there is a chart $\phi :U\to {ℝ}^n$ around it such that $\phi (E\cap U)={ℝ}^k\cap \phi (U).$ Hence, a submanifold is locally closed.

Here is an example in algebraic geometry. Let U be an open affine chart on a projective variety X (in the Zariski topology). Then each closed subvariety Y of U is locally closed in X; namely, $Y=U\cap \bar{Y}$ where $\bar{Y}$ denotes the closure of Y in X. (See also quasi-projective variety and quasi-affine variety.)

Properties

Finite intersections and the pre-image under a continuous map of locally closed sets are locally closed. On the other hand, a union and a complement of locally closed subsets need not be locally closed. (This motivates the notion of a constructible set.)

Especially in stratification theory, for a locally closed subset $E,$ the complement $\bar{E}\setminus E$ is called the boundary of $E$ (not to be confused with topological boundary). If $E$ is a closed submanifold-with-boundary of a manifold $M,$ then the relative interior (that is, interior as a manifold) of $E$ is locally closed in $M$ and the boundary of it as a manifold is the same as the boundary of it as a locally closed subset.

A topological space is said to be submaximal if every subset is locally closed. See Glossary of topology#S for more of this notion.

External links

• locally closed set at the nLab