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.

[The point x is an interior point of S.

The point y is on the boundary of S. | upload.wikimedia.org/wikipedia/commons/b/ba/Interior///illustration.svg]

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 a real number r > 0, such that y is in S whenever the distance This definition generalizes to topological spaces by replacing "open ball" with "open set". If S is a subset of a topological space X then x is an of S in X if x is contained in an open subset of X that 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 or or S^\circ, can be defined in any of the following equivalent ways: If the space X is understood from context then the shorter notation is usually preferred to

Examples

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

Properties

Let X be a topological space and let S and T be subsets of X. Other properties include: Relationship with closure The above statements will remain true if all instances of the symbols/words are respectively replaced by and the following symbols are swapped: For more details on this matter, see interior operator below or the article Kuratowski closure axioms.

Interior operator

The interior operator is dual to the closure operator, which is denoted by or by an overline —, in the sense that and also where X is the topological space containing S, and the backslash 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 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 S of a topological space X, denoted by or simply is the largest open set disjoint from S, namely, it is the union of all open sets in X that are disjoint from S. The exterior is the interior of the complement, which is the same as the complement of the closure; in formulas, Similarly, the interior is the exterior of the complement: The interior, boundary, and exterior of a set S together partition the whole space into three blocks (or fewer when one or more of these is empty): where \partial S denotes the boundary of S. 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:

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.