Exterior (topology)

In topology, the exterior of a subset S of a topological space X is the union of all open sets of X which are disjoint from S. It is itself an open set and is disjoint from S. The exterior of S is denoted by

ext S

Se.

Equivalent definitions

The exterior is equal to X \ S̅, the complement of the topological closure of S and to the interior of the complement of S in X.
Properties

Many properties follow in a straightforward way from those of the interior operator, such as the following.

ext(S) is an open set that is disjoint with S.
ext(S) is the union of all open sets that are disjoint with S.
ext(S) is the largest open set that is disjoint with S.
If S is a subset of T, then ext(S) is a superset of ext(T).

Unlike the interior operator, ext is not idempotent, but the following holds:

ext(ext(S)) is a superset of int(S).