closure

The closure of a set is what you get by adding in every point the set reaches toward but might be missing — you seal up all the gaps along the edge. It is the smallest closed set that still contains the original.

Formally, the closure of A, written A-bar or cl(A), is the union of A with all of its limit points. Equivalently it is the intersection of all closed sets containing A — the smallest such set. A clean test: a point x lies in the closure of A exactly when every open ball around x meets A, no matter how small the ball.

The closure is always closed, always contains A, and equals A precisely when A is already closed. Closure and interior are dual operations linked by complementation: the closure of A is the complement of the interior of the complement of A. The closure can be much larger than A — the closure of the rationals Q in R is all of R.