Interior (topology)

In mathematics, specifically in topology,

the interior of a subset S of a topological space X is the union of all subsets of S that are open in X.

A point that is in the interior of S is an interior point of S.

The interior of S is the complement of the closure of the complement of S.

In this sense interior and closure are dual notions.

The exterior of a set S is the complement of the closure of S; 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

If

S

{\displaystyle S}

is a subset of a Euclidean space, then

x

{\displaystyle x}

is an interior point of

S

{\displaystyle S}

if there exists an open ball centered at

x

{\displaystyle x}

which is completely contained in

S

.

{\displaystyle S.}

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

This definition generalizes to any subset

S

{\displaystyle S}

of a metric space

X

{\displaystyle X}

with metric

d

{\displaystyle d}

:

x

{\displaystyle x}

is an interior point of

S

{\displaystyle S}

if there exists a real number

r

>

0

,

{\displaystyle r>0,}

such that

y

{\displaystyle y}

is in

S

{\displaystyle S}

whenever the distance

d

(

x

,

y

)

<

r

.

{\displaystyle d(x,y)<r.}

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

If

S

{\displaystyle S}

is a subset of a topological space

X

{\displaystyle X}

then

x

{\displaystyle x}

is an interior point of

S

{\displaystyle S}

in

X

{\displaystyle X}

if

x

{\displaystyle x}

is contained in an open subset of

X

{\displaystyle X}

that is completely contained in

S

.

{\displaystyle S.}

(Equivalently,

x

{\displaystyle x}

is an interior point of

S

{\displaystyle S}

if

S

{\displaystyle S}

is a neighbourhood of

x

.

{\displaystyle x.}

)

Interior of a set

The interior of a subset

S

{\displaystyle S}

of a topological space

X

,

{\displaystyle X,}

denoted by

int

X

⁡

S

{\displaystyle \operatorname {int} _{X}S}

or

int

⁡

S

{\displaystyle \operatorname {int} S}

or

S

∘

,

{\displaystyle S^{\circ },}

can be defined in any of the following equivalent ways:

int

⁡

S

{\displaystyle \operatorname {int} S}

is the largest open subset of

X

{\displaystyle X}

contained in

S

.

{\displaystyle S.}

int

⁡

S

{\displaystyle \operatorname {int} S}

is the union of all open sets of

X

{\displaystyle X}

contained in

S

.