PlanetMath: interior

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interior

(Definition)

Let be a subset of a topological space .

The union of all open sets contained in is defined to be the interior of . Equivalently, one could define the interior of to the be the largest open set contained in .

In this entry we denote the interior of by $\operatorname{int}(A)$ . Another common notation is $A^\circ$ .

The exterior of is defined as the union of all open sets whose intersection with is empty. That is, the exterior of is the interior of the complement of .

The interior of a set enjoys many special properties, some of which are listed below:

$\operatorname{int}(A)\subseteq A$
$\operatorname{int}(A)$ is open
$\operatorname{int}(\operatorname{int}(A))=\operatorname{int}(A)$
$\operatorname{int}(X)=X$
$\operatorname{int}(\varnothing )=\varnothing$
is open if and only if $A=\operatorname{int}(A)$
$\overline{A^\complement}=(\operatorname{int}(A))^\complement$
$\overline{A}^\complement = \operatorname{int}(A^\complement)$
$A\subseteq B$ implies that $\operatorname{int}(A)\subseteq \operatorname{int}(B)$
$\operatorname{int}(A)=A\setminus \partial A$ , where $\partial A$ is the boundary of
$X=\operatorname{int}(A)\cup \partial A \cup \operatorname{int}(A^\complement)$

Bibliography

1: S. Willard, General Topology, Addison-Wesley Publishing Company, 1970.

"interior" is owned by yark. [ full author list (3) | owner history (2) ]

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See Also: complement, closure, boundary / frontier

Also defines:

exterior

Keywords:

topology

Attachments:: interior axioms (Definition) by rspuzio
derivation of properties on interior operation (Derivation) by CWoo

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Cross-references: boundary, open, complement, intersection, contained, open sets, union, topological space, subset
There are 98 references to this entry.

This is version 16 of interior, born on 2002-06-21, modified 2008-03-20.
Object id is 3123, canonical name is Interior.
Accessed 11058 times total.

Classification:

AMS MSC:

54-00 (General topology :: General reference works )

Pending Errata and Addenda

None.[ View all 8 ]

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