# 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 $S$ in $X$ is often denoted by $\operatorname {ext} _{X}S$ or, if $X$ is clear from context, then possibly also by $\operatorname {ext} S$ or $S^{\operatorname {e} }.$ ## Equivalent definitions

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

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

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

• $\operatorname {ext} _{X}S$ is an open subset of $X$ that is disjoint from $S.$ • If $S\subseteq T$ then $\operatorname {ext} _{X}T\subseteq \operatorname {ext} _{X}S.$ • $\operatorname {ext} _{X}S$ is equal to the union of all open subsets of $X$ that are disjoint from $S.$ • $\operatorname {ext} _{X}S$ is equal to the largest open subset of $X$ that is disjoint from $S.$ Unlike the interior operator, $\operatorname {ext} _{X}$ is not idempotent, although it does have the following property:

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