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: 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 is closed in B if and only if and that for a subset E and an open subset U,

Examples

The interval 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, 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 around it such that 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, where 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 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.