Interior (topology) explained

In mathematics, specifically in topology,the interior of a subset of a topological space is the union of all subsets of that are open in .A point that is in the interior of is an interior point of .The interior of is the complement of the closure of the complement of .In this sense interior and closure are dual notions.

The exterior of a set is the complement of the closure of ; it consists of the points that are in neither the set nor its boundary.The interior, boundary, and exterior of a subset together partition the whole space into three blocks (or fewer when one or more of these is empty).

Definitions

Interior point

is a subset of a Euclidean space, then

is an interior point of

if there exists an open ball centered at

which is completely contained in

(This is illustrated in the introductory section to this article.)

This definition generalizes to any subset

of a metric space

with metric

is an interior point of

if there exists a real number

r>0,

such that

is in

whenever the distance

d(x,y)<r.

This definition generalizes to topological spaces by replacing "open ball" with "open set".If

is a subset of a topological space

then

is an of

is contained in an open subset of

that is completely contained in

(Equivalently,

is an interior point of

is a neighbourhood of

)

Interior of a set

The interior of a subset

of a topological space

denoted by

\operatorname{int}_XS

\operatorname{int}S

S^\circ,

can be defined in any of the following equivalent ways:

\operatorname{int}S

is the largest open subset of

contained in

\operatorname{int}S

is the union of all open sets of

contained in

\operatorname{int}S

is the set of all interior points of

If the space

is understood from context then the shorter notation

\operatorname{int}S

is usually preferred to

\operatorname{int}_XS.

Examples

In any space, the interior of the empty set is the empty set.
In any space

S\subseteqX,

then

\operatorname{int}S\subseteqS.

is the real line

\Reals

(with the standard topology), then

\operatorname{int}([0,1])=(0,1)

whereas the interior of the set

of rational numbers is empty:

\operatorname{int}\Q=\varnothing.

is the complex plane

\Complex,

then

\operatorname{int}(\{z\in\Complex:|z|\leq1\})=\{z\in\Complex:|z|<1\}.

In any Euclidean space, the interior of any finite set is the empty set.

On the set of real numbers, one can put other topologies rather than the standard one:

is the real numbers

\Reals

with the lower limit topology, then

\operatorname{int}([0,1])=[0,1).

If one considers on

\Reals

the topology in which every set is open, then

\operatorname{int}([0,1])=[0,1].

If one considers on

\Reals

the topology in which the only open sets are the empty set and

\Reals

itself, then

\operatorname{int}([0,1])

is the empty set.

These examples show that the interior of a set depends upon the topology of the underlying space.The last two examples are special cases of the following.

In any discrete space, since every set is open, every set is equal to its interior.

since the only open sets are the empty set and

itself,

\operatorname{int}X=X

and for every proper subset

\operatorname{int}S

is the empty set.

Properties

Let

be a topological space and let

and

be subsets of

\operatorname{int}S

is open in

then

T\subseteqS

if and only if

T\subseteq\operatorname{int}S.

\operatorname{int}S

is an open subset of

when

is given the subspace topology.

is an open subset of

if and only if

\operatorname{int}S=S.

\operatorname{int}S\subseteqS.

\operatorname{int}(\operatorname{int}S)=\operatorname{int}S.

\operatorname{int}(S\capT)=(\operatorname{int}S)\cap(\operatorname{int}T).

- However, the interior operator does not distribute over unions since only

\operatorname{int}(S\cupT)~\supseteq~(\operatorname{int}S)\cup(\operatorname{int}T)

is guaranteed in general and equality might not hold. For example, if

X=\Reals,S=(-infty,0],

and

T=(0,infty)

then

(\operatorname{int}S)\cup(\operatorname{int}T)=(-infty,0)\cup(0,infty)=\Reals\setminus\{0\}

is a proper subset of

\operatorname{int}(S\cupT)=\operatorname{int}\Reals=\Reals.

/: If

S\subseteqT

then

\operatorname{int}S\subseteq\operatorname{int}T.

Other properties include:

is closed in

and

\operatorname{int}T=\varnothing

then

\operatorname{int}(S\cupT)=\operatorname{int}S.

Relationship with closure

The above statements will remain true if all instances of the symbols/words

"interior", "int", "open", "subset", and "largest"are respectively replaced by

"closure", "cl", "closed", "superset", and "smallest"and the following symbols are swapped:

\subseteq

" swapped with "

\supseteq

\cup

" swapped with "

\cap

"For more details on this matter, see interior operator below or the article Kuratowski closure axioms.

Interior operator

The interior operator

\operatorname{int}_X

is dual to the closure operator, which is denoted by

\operatorname{cl}_X

or by an overline ^—, in the sense that

\operatorname_X S = X \setminus \overline

and also

\overline = X \setminus \operatorname_X (X \setminus S),

where

is the topological space containing

and the backslash

\setminus

denotes set-theoretic difference.Therefore, the abstract theory of closure operators and the Kuratowski closure axioms can be readily translated into the language of interior operators, by replacing sets with their complements in

In general, the interior operator does not commute with unions. However, in a complete metric space the following result does hold:

The result above implies that every complete metric space is a Baire space.

Exterior of a set

The exterior of a subset

of a topological space

denoted by

\operatorname{ext}_XS

or simply

\operatorname{ext}S,

is the largest open set disjoint from

namely, it is the union of all open sets in

that are disjoint from

The exterior is the interior of the complement, which is the same as the complement of the closure; in formulas,

\operatorname S = \operatorname(X \setminus S) = X \setminus \overline.

Similarly, the interior is the exterior of the complement: $\operatorname S = \operatorname(X \setminus S).$

The interior, boundary, and exterior of a set

together partition the whole space into three blocks (or fewer when one or more of these is empty):

X = \operatorname S \cup \partial S \cup \operatorname S,

where

\partialS

denotes the boundary of

The interior and exterior are always open, while the boundary is closed.

Some of the properties of the exterior operator are unlike those of the interior operator:

The exterior operator reverses inclusions; if

S\subseteqT,

then

\operatorname{ext}T\subseteq\operatorname{ext}S.

The exterior operator is not idempotent. It does have the property that

\operatorname{int}S\subseteq\operatorname{ext}\left(\operatorname{ext}S\right).

Interior-disjoint shapes

Two shapes

and

are called interior-disjoint if the intersection of their interiors is empty.Interior-disjoint shapes may or may not intersect in their boundary