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). The interior and exterior are always open while the boundary is always closed. Sets with empty interior have been called boundary sets.

Definitions

Interior point

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

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

This definition generalises to topological spaces by replacing “open ball” with “open set”. Let S be a subset of a topological space X. Then x is an interior point of S if x is contained in an open subset of X which is completely contained in S. (Equivalently, x is an interior point of S if S is a neighbourhood of x.)

Interior of a set

The interior of a subset S of a topological space X, denoted by Int S or S°, can be defined in any of the following equivalent ways:

1. Int S is the largest open subset of X contained (as a subset) in S
2. Int S is the union of all open sets of X contained in S
3. Int S is the set of all interior points of S

Examples

• In any space, the interior of the empty set is the empty set.
• In any space X, if S ⊆ X , then int S ⊆ S.
• If X is the Euclidean space {\displaystyle \mathbb {R} } of real numbers, then int() = (0, 1).
• If X is the Euclidean space {\displaystyle \mathbb {R} }, then the interior of the set {\displaystyle \mathbb {Q} } of rational numbers is empty.
• If X is the complex plane {\displaystyle \mathbb {C} =\mathbb {R} ^{2}}, 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 X = {\displaystyle \mathbb {R} }, where {\displaystyle \mathbb {R} } has the lower limit topology, then int() = [0, 1).
• If one considers on {\displaystyle \mathbb {R} } the topology in which every set is open, then int() = .
• 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 int() 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 X, since the only open sets are the empty set and X itself, we have X = int X and for every proper subset S of X, int S is the empty set.

Properties

Let X be a topological space and let S and T be subset of X.

• Int S is open in X.
• If T is open in X then T ⊆ S if and only if T ⊆ Int S.
• Int S is an open subset of S when S is given the subspace topology.
• S is an open subset of X if and only if S = int S.
• Intensive: Int S ⊆ S.
• Idempotence: Int(Int S) = Int S.
• Preserves/distributes over binary intersection: Int (S ∩ T) = (Int S) ∩ (Int T).
• Monotone/nondecreasing with respect to ⊆: If S ⊆ T then Int S ⊆ Int T.

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. “⊆” swapped with “⊇”
2. “∪” swapped with “∩”

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

Other properties include:

• If S is closed in X and Int T = ∅ then Int (S ∪ T) = Int S.

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)}}}

and also

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

where {\displaystyle X} is the topological space containing {\displaystyle S,} and the backslash {\displaystyle \,\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 {\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 (topological) 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 complement of the closure of {\displaystyle S}:

{\displaystyle \operatorname {ext} _{X}S:=X\setminus \operatorname {cl} _{X}S}

although it can be equivalently defined in terms of the interior by:

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

Alternatively, the interior {\displaystyle \operatorname {int} _{X}S} could instead be defined in terms of the exterior by using the set equality

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

As a consequence of this relationship between the interior and exterior, many properties of the exterior {\displaystyle \operatorname {ext} _{X}S} can be readily deduced directly from those of the interior {\displaystyle \operatorname {int} _{X}S} and elementary set identities. Such properties include the following:

• {\displaystyle \operatorname {ext} _{X}S} is an open subset of {\displaystyle X} that is disjoint from {\displaystyle S.}
• If {\displaystyle S\subseteq T} then {\displaystyle \operatorname {ext} _{X}T\subseteq \operatorname {ext} _{X}S.}
• {\displaystyle \operatorname {ext} _{X}S} is equal to the union of all open subsets of {\displaystyle X} that are disjoint from {\displaystyle S.}
• {\displaystyle \operatorname {ext} _{X}S} is equal to the largest open subset of {\displaystyle X} that is disjoint from {\displaystyle S.}

Unlike the interior operator, {\displaystyle \operatorname {ext} _{X}} is not idempotent, although it does have the property that {\displaystyle \operatorname {int} _{X}S\subseteq \operatorname {ext} _{X}\left(\operatorname {ext} _{X}S\right).}

Interior-disjoint shapes

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