The interval is a locally closed subset of For another example, consider the relative interior of a closed disk in 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 .
Recall that, by definition, a submanifold of an -manifold is a subset such that for each point in there is a chart around it such that 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). 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.)