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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 ...

topology
, 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
disjoint
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_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 (X \setminus S) 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(\operatorname_X S\right).


See also

* * * *


Bibliography

* {{Authority control General topology