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.
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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