Partition of Unity

A locally finite collection of nonnegative smooth functions on a topological manifold, subordinate to an open cover, whose pointwise sum equals one; used to glue local constructions into global objects.

A family {φ_i} of continuous (often smooth) functions on a topological space such that for each point only finitely many φ_i are nonzero (local finiteness), each φ_i has support contained in a member of a given open cover, and the sum over i of φ_i equals the constant function 1 everywhere on the space.

A locally finite collection of nonnegative smooth functions on a topological manifold (or domain) subordinate to an open cover whose pointwise sum is identically one; used to localize global constructions by weighting and gluing local data.