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Home > Interior (topology)
In topology, the interior of a set is the union of all open sets contained in it, and contains all interior point s. 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, So.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
- So = 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.
See also: interior algebra.
General topology
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