6.2Paracompact Spaces

Updated 23 Aug 2026Chapter (PDF)

We only cover local finiteness and paracompactness to the extent necessary to generalize the mechanism of partitions of unity to spaces with a condition weaker than compactness. We also briefly state the major metrization theorems (Nagata-Smirnov and Smirnov) and their immediate consequences for manifolds, while blackboxing the highly technical proofs.

[6.2.1]Definition(Paracompactness)#

A space $X$ is said to be paracompact if every open covering $\mathcal{A}$ of $X$ admits a locally finite open refinement $\mathcal{B}$ that covers $X$ .

[6.2.2]Remark(Intuition behind Paracompactness)#

Paracompactness is a natural generalization of compactness. Instead of demanding a finite subcover (which is often too strong of a global condition), we demand a locally finite open refinement. This ensures that while the total number of open sets in the cover might be infinite, locally around any point, the space only “sees” finitely many sets.

As we will see, this is precisely the topological machinery required to generalize partitions of unity. Because the refinement is locally finite, the seemingly infinite sum $\sum \phi_k(x)$ evaluates to a finite sum in some neighborhood of any point $x$ . Thus, the sum is automatically well-defined and continuous for free, without any need for analytic convergence tests.

[6.2.3]Theorem(Paracompact Hausdorff Spaces are Normal)#

Every paracompact Hausdorff space is normal.

[6.2.4]Theorem(Stone's Theorem)#

Every metrizable space is paracompact.

We omit the proofs of these theorems as they are highly technical, but their consequences are immense. Specifically, they allow us to generalize the existence of partitions of unity from compact spaces to any paracompact space.

[6.2.5]Theorem(Existence of Partitions of Unity)#

Let $X$ be a paracompact Hausdorff space, and let $\{U_\alpha\}_{\alpha \in J}$ be an indexed open covering of $X$ . Then there exists a partition of unity $\{\phi_\alpha\}_{\alpha \in J}$ dominated by $\{U_\alpha\}$ .

This theorem is fundamentally why paracompactness is the standard assumption in differential geometry and functional analysis. It guarantees we can take locally defined objects (like coordinate charts, metric tensors, or differential forms) and stitch them together into global objects seamlessly, even if the manifold is not compact.