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
orThe 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 is an open set that is disjoint with S.
- ext is the union of all open sets that are disjoint with S.
- ext is the largest open set that is disjoint with S.
- If S is a subset of T, then ext is a superset of ext.
Unlike the interior operator, ext is not idempotent, but the following holds:
- ext is a superset of int.
