Exterior (topology)

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In topology, the exterior of a subset ${\displaystyle S}$ of a topological space ${\displaystyle X}$ is the union of all open sets of ${\displaystyle X}$ which are disjoint from ${\displaystyle S.}$ It is itself an open set and is disjoint from ${\displaystyle S.}$ The exterior of ${\displaystyle S}$ in ${\displaystyle X}$ is often denoted by ${\displaystyle \operatorname {ext} _{X}S}$ or, if ${\displaystyle X}$ is clear from context, then possibly also by ${\displaystyle \operatorname {ext} S}$ or ${\displaystyle S^{\operatorname {e} }.}$

Equivalent definitions[]

The exterior is equal to ${\displaystyle X\setminus \operatorname {cl} _{X}S,}$ the complement of the (topological) closure of ${\displaystyle S}$ and to the (topological) interior of the complement of ${\displaystyle S}$ in ${\displaystyle X.}$

Properties[]

The topological exterior of a subset ${\displaystyle S\subseteq X}$ always satisfies:

${\displaystyle \operatorname {ext} _{X}S=\operatorname {int} _{X}(X\setminus S)}$

and as a consequence, many properties of ${\displaystyle \operatorname {ext} _{X}S}$ can be readily deduced directly from those of the interior ${\displaystyle \operatorname {int} _{X}S}$ and elementary set identities. Such properties include the following:

• ${\displaystyle \operatorname {ext} _{X}S}$ is an open subset of ${\displaystyle X}$ that is disjoint from ${\displaystyle S.}$
• If ${\displaystyle S\subseteq T}$ then ${\displaystyle \operatorname {ext} _{X}T\subseteq \operatorname {ext} _{X}S.}$
• ${\displaystyle \operatorname {ext} _{X}S}$ is equal to the union of all open subsets of ${\displaystyle X}$ that are disjoint from ${\displaystyle S.}$
• ${\displaystyle \operatorname {ext} _{X}S}$ is equal to the largest open subset of ${\displaystyle X}$ that is disjoint from ${\displaystyle S.}$

Unlike the interior operator, ${\displaystyle \operatorname {ext} _{X}}$ is not idempotent, although it does have the following property:

• ${\displaystyle \operatorname {int} _{X}S\subseteq \operatorname {ext} _{X}\left(\operatorname {ext} _{X}S\right).}$

See also[]

• Closure (topology)
• Boundary (topology)
• Interior (topology) – Largest open subset of some given set
• Jordan curve theorem – Division by a closed curve of the plane into two regions

Bibliography[]

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

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