In topology, a branch of mathematics, a subset
is said to be locally closed if any of the following equivalent conditions are satisfied:[1][2][3][4] 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
and an open subset
is a locally closed subset of
of a closed disk in
It is locally closed since it is an intersection of the closed disk and an open ball.
is not a locally closed subset of
Recall that, by definition, a submanifold
is a subset such that for each point
Hence, a submanifold is locally closed.
[5] Here is an example in algebraic geometry.
Let U be an open affine chart on a projective variety X (in the Zariski topology).
Finite intersections and the pre-image under a continuous map of locally closed sets are locally closed.
[1] On the other hand, a union and a complement of locally closed subsets need not be locally closed.
[6] (This motivates the notion of a constructible set.)
Especially in stratification theory, for a locally closed subset
(not to be confused with topological boundary).
is a closed submanifold-with-boundary of a manifold
is locally closed in
and the boundary of it as a manifold is the same as the boundary of it as a locally closed subset.
[2] A topological space is said to be submaximal if every subset is locally closed.
See Glossary of topology#S for more of this notion.