Interior (topology)

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In mathematics, the interior of a set S consists of all points of S that are intuitively "not on the edge of S". A point that is in the interior of S is an interior point of S.

The exterior of a set is the interior of its complement; it consists of the points that are not in the set or its boundary.

The notion of the interior of a set is a topological concept; it is not defined for all sets, but it is defined for sets that are a subset of a topological space. It is in many ways dual to the notion of closure. In particular the two concepts are dual in the sense of category theory.

Contents

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 contained in S.

This definition generalizes to any subset S of a metric space X. Fully expressed, if X is a metric space with metric d, then 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 "neighbourhood". Let S be a subset of a topological space X. Then x is an interior point of S if there exists a neighbourhood of x which is contained in S. Note that this definition does not depend upon whether neighbourhoods are required to be open. If neighbourhoods are not required to be open then S will automatically be a neighbourhood of x if S contains a neighbourhood of x.

Interior of a set

The interior of a set S is the set of all interior points of S. The interior of S is denoted int(S), Int(S), or So. The interior of a set has the following properties.

  • int(S) is an open subset of S.
  • int(S) is the union of all open sets contained in S.
  • int(S) is the largest open set contained in S.
  • A set S is open if and only if S = int(S).
  • int(int(S)) = int(S) (idempotence).
  • If S is a subset of T, then int(S) is a subset of int(T).
  • If A is an open set, then A is a subset of S if and only if A is a subset of int(S).

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