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).

The interior and exterior of a closed curve are a slightly different concept; see the Jordan curve theorem.

Definitions

Interior point

If ${\displaystyle S}$ is a subset of a Euclidean space, then ${\displaystyle x}$ is an interior point of ${\displaystyle S}$ if there exists an open ball centered at ${\displaystyle x}$ which is completely contained in ${\displaystyle S.}$ (This is illustrated in the introductory section to this article.)

This definition generalizes to any subset ${\displaystyle S}$ of a metric space ${\displaystyle X}$ with metric ${\displaystyle d}$: ${\displaystyle x}$ is an interior point of ${\displaystyle S}$ if there exists a real number ${\displaystyle r>0,}$ such that ${\displaystyle y}$ is in ${\displaystyle S}$ whenever the distance ${\displaystyle d(x,y)<r.}$

This definition generalizes to topological spaces by replacing "open ball" with "open set". If ${\displaystyle S}$ is a subset of a topological space ${\displaystyle X}$ then ${\displaystyle x}$ is an interior point of ${\displaystyle S}$ in ${\displaystyle X}$ if ${\displaystyle x}$ is contained in an open subset of ${\displaystyle X}$ that is completely contained in ${\displaystyle S.}$ (Equivalently, ${\displaystyle x}$ is an interior point of ${\displaystyle S}$ if ${\displaystyle S}$ is a neighbourhood of ${\displaystyle x.}$)

Interior of a set

The interior of a subset ${\displaystyle S}$ of a topological space ${\displaystyle X,}$ denoted by ${\displaystyle \operatorname {int} _{X}S}$ or ${\displaystyle \operatorname {int} S}$ or ${\displaystyle S^{\circ },}$ can be defined in any of the following equivalent ways:

1. ${\displaystyle \operatorname {int} S}$ is the largest open subset of ${\displaystyle X}$ contained in ${\displaystyle S.}$
2. ${\displaystyle \operatorname {int} S}$ is the union of all open sets of ${\displaystyle X}$ contained in ${\displaystyle S.}$
3. ${\displaystyle \operatorname {int} S}$ is the set of all interior points of ${\displaystyle S.}$

If the space ${\displaystyle X}$ is understood from context then the shorter notation ${\displaystyle \operatorname {int} S}$ is usually preferred to ${\displaystyle \operatorname {int} _{X}S.}$

Examples

• In any space, the interior of the empty set is the empty set.
• In any space ${\displaystyle X,}$ if ${\displaystyle S\subseteq X,}$ then ${\displaystyle \operatorname {int} S\subseteq S.}$
• If ${\displaystyle X}$ is the real line ${\displaystyle \mathbb {R} }$ (with the standard topology), then ${\displaystyle \operatorname {int} ([0,1])=(0,1)}$ whereas the interior of the set ${\displaystyle \mathbb {Q} }$ of rational numbers is empty: ${\displaystyle \operatorname {int} \mathbb {Q} =\varnothing }$
• If ${\displaystyle X}$ is the complex plane ${\displaystyle \mathbb {C} ,}$ then ${\displaystyle \operatorname {int} (\{z\in \mathbb {C} :|z|\leq 1\})=\{z\in \mathbb {C} :|z|<1\}.}$
• 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:

• If ${\displaystyle X}$ is the real numbers ${\displaystyle \mathbb {R} }$ with the lower limit topology, then ${\displaystyle \operatorname {int} ([0,1])=[0,1).}$
• If one considers on ${\displaystyle \mathbb {R} }$ the topology in which every set is open, then ${\displaystyle \operatorname {int} ([0,1])=[0,1].}$
• If one considers on ${\displaystyle \mathbb {R} }$ the topology in which the only open sets are the empty set and ${\displaystyle \mathbb {R} }$ itself, then ${\displaystyle \operatorname {int} ([0,1])}$ 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.
• In any indiscrete space ${\displaystyle X,}$ since the only open sets are the empty set and ${\displaystyle X}$ itself, ${\displaystyle \operatorname {int} X=X}$ and for every proper subset ${\displaystyle S}$ of ${\displaystyle X,}$ ${\displaystyle \operatorname {int} S}$ is the empty set.

Properties

Let ${\displaystyle X}$ be a topological space and let ${\displaystyle S}$ and ${\displaystyle T}$ be subsets of ${\displaystyle X.}$

• ${\displaystyle \operatorname {int} S}$ is open in ${\displaystyle X.}$
• If ${\displaystyle T}$ is open in ${\displaystyle X}$ then ${\displaystyle T\subseteq S}$ if and only if ${\displaystyle T\subseteq \operatorname {int} S.}$
• ${\displaystyle \operatorname {int} S}$ is an open subset of ${\displaystyle S}$ when ${\displaystyle S}$ is given the subspace topology.
• ${\displaystyle S}$ is an open subset of ${\displaystyle X}$ if and only if ${\displaystyle \operatorname {int} S=S.}$
• Intensive: ${\displaystyle \operatorname {int} S\subseteq S.}$
• Idempotence: ${\displaystyle \operatorname {int} (\operatorname {int} S)=\operatorname {int} S.}$
• Preserves/distributes over binary intersection: ${\displaystyle \operatorname {int} (S\cap T)=(\operatorname {int} S)\cap (\operatorname {int} T).}$
  • However, the interior operator does not distribute over unions since only ${\displaystyle \operatorname {int} (S\cup T)~\supseteq ~(\operatorname {int} S)\cup (\operatorname {int} T)}$ is guaranteed in general and equality might not hold.^{[note 1]} For example, if ${\displaystyle X=\mathbb {R} ,S=(-\infty ,0],}$ and ${\displaystyle T=(0,\infty )}$ then ${\displaystyle (\operatorname {int} S)\cup (\operatorname {int} T)=(-\infty ,0)\cup (0,\infty )=\mathbb {R} \setminus \{0\))$ is a proper subset of ${\displaystyle \operatorname {int} (S\cup T)=\operatorname {int} \mathbb {R} =\mathbb {R} .}$
• Monotone/nondecreasing with respect to ${\displaystyle \subseteq }$: If ${\displaystyle S\subseteq T}$ then ${\displaystyle \operatorname {int} S\subseteq \operatorname {int} T.}$

Other properties include:

• If ${\displaystyle S}$ is closed in ${\displaystyle X}$ and ${\displaystyle \operatorname {int} T=\varnothing }$ then ${\displaystyle \operatorname {int} (S\cup T)=\operatorname {int} S.}$

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:

1. "${\displaystyle \subseteq }$" swapped with "${\displaystyle \supseteq }$"
2. "${\displaystyle \cup }$" swapped with "${\displaystyle \cap }$"

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

Interior operator

The interior operator ${\displaystyle \operatorname {int} _{X))$ is dual to the closure operator, which is denoted by ${\displaystyle \operatorname {cl} _{X))$ or by an overline ^—, in the sense that

$${\displaystyle \operatorname {int} _{X}S=X\setminus {\overline {(X\setminus S)))}$$

$${\displaystyle {\overline {S))=X\setminus \operatorname {int} _{X}(X\setminus S),}$$

${\displaystyle X}$

${\displaystyle S,}$

${\displaystyle \,\setminus \,}$

${\displaystyle X.}$

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 ${\displaystyle S}$ of a topological space ${\displaystyle X,}$ denoted by ${\displaystyle \operatorname {ext} _{X}S}$ or simply ${\displaystyle \operatorname {ext} S,}$ is the largest open set disjoint from ${\displaystyle S,}$ namely, it is the union of all open sets in ${\displaystyle X}$ that are disjoint from ${\displaystyle S.}$ The exterior is the interior of the complement, which is the same as the complement of the closure;^[2] in formulas,

$${\displaystyle \operatorname {ext} S=\operatorname {int} (X\setminus S)=X\setminus {\overline {S)).}$$

Similarly, the interior is the exterior of the complement:

$${\displaystyle \operatorname {int} S=\operatorname {ext} (X\setminus S).}$$

The interior, boundary, and exterior of a set ${\displaystyle S}$ together partition the whole space into three blocks (or fewer when one or more of these is empty):

$${\displaystyle X=\operatorname {int} S\cup \partial S\cup \operatorname {ext} S,}$$

${\displaystyle \partial S}$

${\displaystyle S.}$

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

• The exterior operator reverses inclusions; if ${\displaystyle S\subseteq T,}$ then ${\displaystyle \operatorname {ext} T\subseteq \operatorname {ext} S.}$
• The exterior operator is not idempotent. It does have the property that ${\displaystyle \operatorname {int} S\subseteq \operatorname {ext} \left(\operatorname {ext} S\right).}$

Interior-disjoint shapes

Two shapes ${\displaystyle a}$ and ${\displaystyle b}$ are called interior-disjoint if the intersection of their interiors is empty. Interior-disjoint shapes may or may not intersect in their boundary.