Definition
Closure
The closure of a set
, denoted by , is the set of all adherent points of . A point is an adherent point of if every neighbourhood of contains at least one point of . Equivalently, the closure is the union of the interior and the boundary of the set:
.
Properties
- Closedness: The closure
is always a closed set. In fact, it is the smallest closed set containing . - Idempotency: Taking the closure of a closure does not change the set:
. - Containment:
. A set is closed if and only if . - Limit Points: In a metric space, the closure of a set consists of the set itself plus all of its limit points.