# Exterior (topology)

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:

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