Interior topology

In topology, the interior of a set is the union of all open sets contained in it, and contains all interior points. It is the largest open set contained in the original set. The interior of a set S is denoted by int S, Int S, or, S^o.

A set S is an open set if and only if S is equal to the interior of S.

Examples

The interior of the empty set is empty.
int [1, 10] = (1, 10)
int {(x, y) ∈ R² | y ∈ (0, 1]} = {(x, y) ∈ R² | y ∈ (0, 1)}

The interior operator ^o is dual to the closure operator ^-, in the sense that

S^o = X \ (X \ S)^-,

and also

S^- = X \ (X \ S)^o.

Therefore the abstract theory of closure operators and Kuratowski closure operators reads easily across to the analogues for interiors, by replacing sets with their complements.