= Closure = A '''closure''' of a [[Analysis/Sets|set]] is the union of the set with all of its [[Analysis/LimitPoint|limit points]]. <> ---- == Description == Openness of a set can be determined in either a [[Analysis/TopologicalSpace|topological space]] or a [[Analysis/MetricSpace|metric space]]. The core idea is that a set is closed if it contains all of its [[Analysis/LimitPoint|limit points]], and open if it contains none of them. The union of a set with all of its [[Analysis/LimitPoint|limit points]] is defined as its closure. {{attachment:closure.svg}} In circumstances where it isn't clear what [[Analysis/TopologicalSpace|topological space]] ''A'' is a subset of, the most common notation is ''cl,,(X, τ),,(A)''. A closure is always closed. The statements that 'a set is equal to its own closure' and that 'a set is closed' are equivalent. ---- CategoryRicottone