interior

The interior of a set is its solid core: the points sitting comfortably inside, each surrounded by a little room that is still entirely within the set. Anything pressed up against the edge, with the set's complement infinitely close, is excluded.

Formally, a point x is an interior point of a set A if some open ball B(x, r) is contained in A. The interior of A, written int(A) or A-with-a-small-circle, is the set of all such points. It can be characterized two ways at once: it is the largest open set contained in A, and equivalently it is the union of all open subsets of A.

The interior is always open, always contained in A, and equals A precisely when A is open. Taking the interior can shrink a set dramatically: the interior of the rationals Q inside R is empty, because every open ball on the line contains irrational numbers, so no rational has room to spare.