partitionofunity

Partition of unity is a mathematical construction used in topology and analysis to assemble global objects from local data. Given a topological space X and an open cover {U_i} indexed by I, a partition of unity subordinate to this cover is a collection of nonnegative functions {φ_i: X → [0, ∞)} such that:

- each φ_i is supported in U_i (i.e., φ_i(x) = 0 for x ∉ U_i),

- the family {φ_i} is locally finite (at every x, only finitely many φ_i(x) are nonzero),

- for every x in X, the sum over all i of φ_i(x) equals 1.

If X is a smooth manifold and each φ_i is smooth, the collection is a smooth partition of unity.

Existence and basic properties: On paracompact spaces, every open cover admits a partition of unity subordinate to it. Consequently, every smooth manifold admits smooth partitions of unity subordinate to any open cover. Partitions of unity are typically not unique; they can be refined, scaled, or combined to suit a particular construction.

Uses and significance: Partitions of unity enable the globalization of local constructions. They allow gluing of local functions or sections of bundles into global objects, the definition of global Riemannian metrics from local data, and the extension of functions from closed subsets to the whole space. They also underpin many proofs in differential geometry, analysis on manifolds, and topology, including the Tietze extension theorem on normal spaces and various approximation arguments.

See also: paracompactness, Urysohn's lemma, bump functions, smooth partitions of unity, differential geometry.