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 point x is an interior point of S. The point y is on the boundary of S.

1 Definitions

1.1 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.)

1.2 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:

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

2 Examples

3 Properties

4 Interior operator

5 Exterior of a set

6 Interior-disjoint shapes

7 See also