TheInfoList

In
topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

, the exterior of a subset $S$ of a
topological space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
$X$ is the union of all
open set In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s of $X$ which are from $S.$ It is itself an open set and is disjoint from $S.$ The exterior of $S$ in $X$ is often denoted by $\operatorname_X S$ or, if $X$ is clear from context, then possibly also by $\operatorname S$ or $S^.$

# Equivalent definitions

The exterior is equal to $X \setminus \operatorname_X S,$ the
complement A complement is often something that completes something else, or at least adds to it in some useful way. Thus it may be: * Complement (linguistics), a word or phrase having a particular syntactic role ** Subject complement, a word or phrase addi ...
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: :$\operatorname_X S = \operatorname_X \left(X \setminus S\right)$ and as a consequence, many properties of $\operatorname_X S$ can be readily deduced directly from those of the
interior Interior may refer to: Arts and media * Interior (Degas), ''Interior'' (Degas) (also known as ''The Rape''), painting by Edgar Degas * Interior (play), ''Interior'' (play), 1895 play by Belgian playwright Maurice Maeterlinck * The Interior (novel) ...
$\operatorname_X S$ and elementary set identities. Such properties include the following: * $\operatorname_X S$ is an open subset of $X$ that is disjoint from $S.$ * If $S \subseteq T$ then $\operatorname_X T \subseteq \operatorname_X S.$ * $\operatorname_X S$ is equal to the union of all open subsets of $X$ that are disjoint from $S.$ * $\operatorname_X S$ is equal to the largest open subset of $X$ that is disjoint from $S.$ Unlike the interior operator, $\operatorname_X$ is not
idempotent Idempotence (, ) is the property of certain operations Operation or Operations may refer to: Science and technology * Surgical operation Surgery ''cheirourgikē'' (composed of χείρ, "hand", and ἔργον, "work"), via la, chirurgiae, ...
, although it does have the following property: * $\operatorname_X S \subseteq \operatorname_X \left\left(\operatorname_X S\right\right).$