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 $$\overline{E}.$$
 * The set $$\overline{E}\setminus E$$ 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 $$A = \overline{A} \cap B$$ and that for a subset $$E$$ and an open subset $$U,$$ $$\overline{E} \cap U = \overline{E \cap U} \cap U.$$

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

On the other hand, $$\{ (x,y)\in\Reals^2 \mid x\ne0 \} \cup \{(0,0)\}$$ is not a locally closed subset of $$\Reals^2$$.

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 $$\varphi : U \to \Reals^n$$ around it such that $$\varphi(E \cap U) = \Reals^k \cap \varphi(U).$$ 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, $$Y = U \cap \overline{Y}$$ where $$\overline{Y}$$ 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 $$\overline{E} \setminus E$$ 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  if every subset is locally closed. See Glossary of topology for more of this notion.