Exterior (topology)

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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[edit]

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

Properties[edit]

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[edit]