Every open cover of a smooth paracompact manifold admits a smooth partition of unity subordinate to it: nonnegative smooth functions with locally finite supports in the cover whose sum is one.
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A **partition of unity** is a mathematical concept used in various fields such as analysis, topology, and differential geometry. It refers to a collection of continuous functions that are used to locally "patch together" global constructs, such as functions or forms, in a coherent way. ### Definition: Let \( M \) be a topological space (often a manifold).