Paracompact manifold
A smooth manifold whose underlying topological space is paracompact, enabling global constructions via partitions of unity.
A smooth manifold is called paracompact if its underlying topological space is paracompact (and, in the usual manifold conventions, Hausdorff).
Global constructions
Paracompactness is the standard condition that guarantees the existence of smooth partitions of unity and allows one to globalize local data (metrics, connections, differential forms defined on charts, and so on).
Key consequence
On a paracompact manifold, every open cover admits a smooth partition of unity subordinate to that cover.
In many texts, paracompactness is built into the definition of a smooth manifold (often via second countability), precisely to ensure this consequence.
Examples
- Euclidean spaces and standard manifolds. , spheres, tori, and any Hausdorff manifold with a countable atlas are paracompact.
- Compact manifolds. Every compact manifold is paracompact (compactness implies paracompactness at the topological level).
- A standard nonexample (topological). If second countability is omitted from the manifold convention, the long line is a Hausdorff locally Euclidean one-dimensional space that is not second countable and is not paracompact; it illustrates why paracompactness (or a countability hypothesis) is not automatic for all manifolds.