Paracompactness is the topological condition that lets local constructions be patched globally. Partitions of unity are its analytic expression.
Locally finite refinements
Local finiteness prevents infinitely many sets from interacting near a point.
Definition. A family \(\mathcal V\) of subsets of \(X\) is locally finite if every point has a neighborhood meeting only finitely many members of \(\mathcal V\). A space is paracompact if every open cover has a locally finite open refinement.
Proposition. In a locally finite family \(\{A_i\}\), \(\overline{\bigcup_iA_i}=\bigcup_i\overline{A_i}\).
Proof. The inclusion from right to left is always true. For the reverse inclusion, if \(x\notin\bigcup_i\overline{A_i}\), choose a neighborhood \(U\) of \(x\) meeting only finitely many \(A_i\). Shrink \(U\) further to avoid the finitely many relevant closures. The resulting neighborhood misses every \(A_i\), so \(x\notin\overline{\bigcup_iA_i}\).
Paracompact Hausdorff spaces
Paracompactness plus Hausdorff separation implies strong normality properties.
Theorem. Every paracompact Hausdorff space is normal.
Proof. Let \(A,B\) be disjoint closed sets. For each \(a\in A\), Hausdorff regularity in a paracompact Hausdorff space gives an open neighborhood \(U_a\) whose closure misses \(B\). Refine the cover \(\{U_a:a\in A\}\cup\{X\setminus A\}\) by a locally finite open cover. The union of refinement elements meeting \(A\) is an open neighborhood of \(A\); local finiteness ensures its closure is contained in the union of the corresponding closures and still misses \(B\). Symmetrically obtain an open neighborhood of \(B\) disjoint from the first.
Remark. The proof shows why local finiteness is the right finiteness condition: it allows closures to commute with unions.
Partitions of unity
A partition of unity turns an open cover into continuous weights.
Definition. A partition of unity subordinate to an open cover \(\mathcal U\) is a family of continuous functions \(\{\varphi_i:X\to[0,1]\}\) such that the supports \(\operatorname{supp}\varphi_i=\overline{\{x:\varphi_i(x)\ne0\}}\) are locally finite, each support lies in some member of \(\mathcal U\), and \(\sum_i\varphi_i(x)=1\) for all \(x\).
Theorem. Every open cover of a paracompact Hausdorff space admits a subordinate partition of unity.
Proof. First take a locally finite open refinement and then shrink it so that closures remain inside chosen cover elements, using normality. Urysohn functions produce bump functions that are positive on the smaller sets and supported in the larger sets. Local finiteness makes the pointwise sum finite and continuous. Dividing each bump function by the total sum gives a partition of unity.
Metric spaces
Metric spaces are paracompact, which explains why partitions of unity are so common in analysis and geometry.
Theorem. Every metric space is paracompact.
Proof. Given an open cover, assign to each point a radius small enough that its ball lies in some cover element. For each scale \(n\), choose a maximal separated set among points with radius at least \(2^{-n}\), and take appropriately sized balls around those chosen points. The separatedness gives local finiteness at each scale, and the scale construction yields a \(\sigma\)-locally finite open refinement. A standard collection-by-first-scale argument converts this into a locally finite refinement.
Remark. This theorem is deep enough that many notes use it as a black box. The key geometric idea is that metric balls can be chosen with controlled overlap.
From refinements to global constructions
Theorem. Every paracompact Hausdorff space is normal. Every open cover of a paracompact Hausdorff space admits a locally finite partition of unity subordinate to it when the space is a smooth manifold, and a continuous partition under standard complete regularity hypotheses.
Local finiteness is the crucial condition: at each point only finitely many summands are nonzero, so \[ \sum_i\rho_i=1 \] is continuous or smooth without any convergence problem. If local data \(T_i\) live in a convex space, then \[ T=\sum_i\rho_iT_i \] patches them globally while retaining the desired local property.
Example (Riemannian metrics). Pull the Euclidean metric back through each manifold chart to obtain \(g_i\). A subordinate smooth partition gives \[ g=\sum_i\rho_i g_i. \] Every \(g_i\) is positive definite and the coefficients are nonnegative with sum one, so \(g\) is a global Riemannian metric. The same method constructs connections, bundle metrics, and globally defined test functions.
Counterexample. The long line is locally Euclidean and Hausdorff but not paracompact. Thus local manifold-like behavior alone does not guarantee partitions of unity; second countability in the standard manifold definition supplies paracompactness.