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
or
- 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).
See also
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