Exterior (topology)
In topology, the exterior of a subset [math]\displaystyle{ S }[/math] of a topological space [math]\displaystyle{ X }[/math] is the union of all open sets of [math]\displaystyle{ X }[/math] which are disjoint from [math]\displaystyle{ S. }[/math] It is itself an open set and is disjoint from [math]\displaystyle{ S. }[/math] The exterior of [math]\displaystyle{ S }[/math] in [math]\displaystyle{ X }[/math] is often denoted by [math]\displaystyle{ \operatorname{ext}_X S }[/math] or, if [math]\displaystyle{ X }[/math] is clear from context, then possibly also by [math]\displaystyle{ \operatorname{ext} S }[/math] or [math]\displaystyle{ S^{\operatorname{e}}. }[/math]
Equivalent definitions
The exterior is equal to [math]\displaystyle{ X \setminus \operatorname{cl}_X S, }[/math] the complement of the (topological) closure of [math]\displaystyle{ S }[/math] and to the (topological) interior of the complement of [math]\displaystyle{ S }[/math] in [math]\displaystyle{ X. }[/math]
Properties
The topological exterior of a subset [math]\displaystyle{ S \subseteq X }[/math] always satisfies:
- [math]\displaystyle{ \operatorname{ext}_X S = \operatorname{int}_X (X \setminus S) }[/math]
and as a consequence, many properties of [math]\displaystyle{ \operatorname{ext}_X S }[/math] can be readily deduced directly from those of the interior [math]\displaystyle{ \operatorname{int}_X S }[/math] and elementary set identities. Such properties include the following:
- [math]\displaystyle{ \operatorname{ext}_X S }[/math] is an open subset of [math]\displaystyle{ X }[/math] that is disjoint from [math]\displaystyle{ S. }[/math]
- If [math]\displaystyle{ S \subseteq T }[/math] then [math]\displaystyle{ \operatorname{ext}_X T \subseteq \operatorname{ext}_X S. }[/math]
- [math]\displaystyle{ \operatorname{ext}_X S }[/math] is equal to the union of all open subsets of [math]\displaystyle{ X }[/math] that are disjoint from [math]\displaystyle{ S. }[/math]
- [math]\displaystyle{ \operatorname{ext}_X S }[/math] is equal to the largest open subset of [math]\displaystyle{ X }[/math] that is disjoint from [math]\displaystyle{ S. }[/math]
Unlike the interior operator, [math]\displaystyle{ \operatorname{ext}_X }[/math] is not idempotent, although it does have the following property:
- [math]\displaystyle{ \operatorname{int}_X S \subseteq \operatorname{ext}_X \left(\operatorname{ext}_X S\right). }[/math]
See also
- Closure (topology) – All points and limit points in a subset of a topological space
- Boundary (topology) – All points not part of the interior of a subset of a topological space
- Interior (topology) – Largest open subset of some given set
- Jordan curve theorem – A closed curve divides the plane into two regions
Bibliography
- Willard, Stephen (2004). General Topology. Dover Books on Mathematics (First ed.). Mineola, N.Y.: Dover Publications. ISBN 978-0-486-43479-7. OCLC 115240.
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