Exterior (topology)

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 definitionsEdit

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

PropertiesEdit

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 alsoEdit

BibliographyEdit

  • 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.