Compactness
Compactness [ edit ] Main article: Compactness Compactness is a concept from general topology that plays an important role in many of the theorems of real analysis. The property of compactness is a generalization of the notion of a set being closed and bounded . (In the context of real analysis, these notions are equivalent: a set in Euclidean space is compact if and only if it is closed and bounded.) Briefly, a closed set contains all of its boundary points , while a set is bounded if there exists a real number such that the distance between any two points of the set is less than that number. In � , sets that are closed and bounded, and therefore compact, include the empty set, any finite number of points, closed intervals , and their finite unions. However, this list is not exhaustive; for instance, the set { 1 / � : � ∈ � } ∪ { 0 } is a compact set; the Cantor ternary set � ⊂...