Exterior (topology)

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In topology, the exterior of a subset of a topological space is the union of all open sets of which are disjoint from It is itself an open set and is disjoint from The exterior of in is often denoted by or, if is clear from context, then possibly also by or

Equivalent definitions[edit]

The exterior is equal to the complement of the (topological) closure of and to the (topological) interior of the complement of in

Properties[edit]

The topological exterior of a subset always satisfies:

and as a consequence, many properties of can be readily deduced directly from those of the interior and elementary set identities. Such properties include the following:

  • is an open subset of that is disjoint from
  • If then
  • is equal to the union of all open subsets of that are disjoint from
  • is equal to the largest open subset of that is disjoint from

Unlike the interior operator, is not idempotent, although it does have the following property:

See also[edit]

Bibliography[edit]

  • Willard, Stephen (2004) [1970]. General Topology. Dover Books on Mathematics (First ed.). Mineola, N.Y.: Dover Publications. ISBN 978-0-486-43479-7. OCLC 115240.