Überdeckung

Überdeckung, literally meaning covering, is a fundamental concept in set theory and topology. Let X be a set and U = {U_i}_{i∈I} a family of subsets of X. U is called an Überdeckung (covering) of X if X ⊆ ∪_{i∈I} U_i. If each U_i is open in a given topology on X, the covering is called an open cover. A subcollection V ⊆ U is a subcover if X ⊆ ∪_{V∈V} V; in particular, a finite subcover consists of finitely many sets from U that still cover X.

Compactness is closely tied to coverings: a topological space X is compact if every open cover of X has a finite subcover. This property characterizes many important behaviors and underpins results such as the existence of finite subcovers for open coverings in the compact setting. In metric spaces, a related concept is the Lebesgue number: if X is compact and U is an open cover, there exists δ > 0 such that every subset of X with diameter less than δ is contained in some member of U.

Refinements and nerves provide structural tools for coverings. A cover V is a refinement of U if every V_j ∈ V is contained in some U_i ∈ U. The nerve of a cover, an object studied in algebraic topology, captures the intersection pattern of the covering sets.

Examples help illustrate the idea: for X = {1,2,3}, the family {{1,2}, {2,3}} is an Überdeckung of X since their union is {1,2,3}. The cover {{1}, {2}} does not cover X. Covers are used to study properties of spaces, to construct partitions of unity, and to define notions such as compactness and various covering-related invariants.