Paracompactness

Paracompactness is a covering property that generalizes compactness and plays a decisive role in the Nagata-Smirnov metrization theorem. Every metrizable space is paracompact, and paracompact Hausdorff spaces enjoy many of the strong properties of compact spaces.

Definitions

Definition7.7Refinement

Let $\mathcal{U}$ and $\mathcal{V}$ be covers of a space $X$ . We say $\mathcal{V}$ is a refinement of $\mathcal{U}$ if every element of $\mathcal{V}$ is contained in some element of $\mathcal{U}$ : for every $V \in \mathcal{V}$ , there exists $U \in \mathcal{U}$ with $V \subseteq U$ .

Definition7.8Locally Finite Family

A family $\{A_\alpha\}_{\alpha \in I}$ of subsets of $X$ is locally finite if every point $x \in X$ has a neighborhood $U$ that intersects only finitely many $A_\alpha$ : $|\{\alpha \in I : U \cap A_\alpha \neq \emptyset\}| < \infty.$

Definition7.9Paracompact Space

A topological space $X$ is paracompact if every open cover has an open locally finite refinement. That is, for every open cover $\mathcal{U}$ , there exists an open cover $\mathcal{V}$ that is a refinement of $\mathcal{U}$ and is locally finite.

Basic Properties

Theorem7.6Properties of Paracompact Hausdorff Spaces

Every compact space is paracompact.
Every metrizable space is paracompact.
Every paracompact Hausdorff space is normal.
Every closed subspace of a paracompact space is paracompact.
A paracompact Hausdorff space is regular.

Proof

(1): A finite subcover is trivially locally finite.

(3): Let $A, B$ be disjoint closed sets in the paracompact Hausdorff space $X$ . For each $x \in X$ , choose an open $U_x$ such that $\overline{U_x}$ is disjoint from $A$ or from $B$ (possible by regularity, which we get from Hausdorff + paracompact $\Rightarrow$ regular). The details use a locally finite refinement and the shrinking lemma.

More precisely: for each $a \in A$ , since $X$ is Hausdorff and hence regular (this follows from paracompact + $T_2$ ), there is an open $U_a$ with $a \in U_a \subseteq \overline{U_a} \subseteq X \setminus B$ . The family $\{U_a : a \in A\} \cup \{X \setminus A\}$ is an open cover. Take a locally finite refinement $\mathcal{V}$ . Then $\bigcup\{V \in \mathcal{V} : V \cap A \neq \emptyset\}$ is an open set containing $A$ with closure disjoint from $B$ .

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Examples

ExampleParacompact Spaces

$\mathbb{R}^n$ : Metrizable, hence paracompact.
All manifolds (by convention second-countable Hausdorff, hence metrizable, hence paracompact).
CW complexes: Paracompact (but the proof is nontrivial).
$[0, \omega_1)$ (first uncountable ordinal): Paracompact but not compact.
Any discrete space: Paracompact (every cover is already locally finite).

ExampleNon-Paracompact Spaces

The Sorgenfrey plane $\mathbb{R}_\ell \times \mathbb{R}_\ell$ : Not paracompact (not even normal).
The long line: Not paracompact (not metrizable, and fails to have locally finite refinements for certain covers).

Partitions of Unity

Definition7.10Partition of Unity

Let $X$ be a topological space and $\mathcal{U} = \{U_\alpha\}$ an open cover. A partition of unity subordinate to $\mathcal{U}$ is a family of continuous functions $\{\phi_\alpha: X \to [0, 1]\}$ such that:

$\operatorname{supp}(\phi_\alpha) \subseteq U_\alpha$ for each $\alpha$ .
$\{\operatorname{supp}(\phi_\alpha)\}$ is locally finite.
$\sum_\alpha \phi_\alpha(x) = 1$ for all $x \in X$ .

(The sum in (3) is well-defined since only finitely many terms are nonzero near any point.)

Theorem7.7Existence of Partitions of Unity

A Hausdorff space $X$ is paracompact if and only if every open cover admits a partition of unity subordinate to it.

RemarkApplications of Partitions of Unity

Partitions of unity are used extensively in:

Differential geometry: Constructing global objects (metrics, connections, differential forms) from local data.
Analysis: Proving extension theorems, localizing problems.
Algebraic topology: The existence of partitions of unity on paracompact spaces implies that Cech cohomology and sheaf cohomology agree.

Sigma-Locally Finite Bases

Definition7.11$\sigma$-Locally Finite Family

A family $\mathcal{A}$ of subsets is $\sigma$ -locally finite if $\mathcal{A} = \bigcup_{n=1}^{\infty} \mathcal{A}_n$ where each $\mathcal{A}_n$ is locally finite.

RemarkRole in Metrization

The Nagata-Smirnov metrization theorem states that a topological space is metrizable if and only if it is $T_3$ (regular Hausdorff) and has a $\sigma$ -locally finite basis. This is the most general metrization theorem, and the concept of $\sigma$ -locally finite families is its core.