1 Partition of Unity

Definition 1.1 (Support) The support of a function \(f: X → ℝ\) of a topological space \(X\) is the closure of the set

\[ \{x ∈ X: f(x) ≠ 0\} \]

relative to \(X\).

Remark. Always remember the support of a function is a closure taken relative to its domain of definition.

Definition 1.2 (Locally finite) A collection of subsets \(\mathcal A\) of a topological space \(X\) is said to be locally finite in \(X\) if

  • all \(x ∈ X\) has a neighborhood intersecting only finite many sets in \(\mathcal A\).

Definition 1.3 (Partition of unity) A collection of smooth functions \(\mathcal R = \{ρ: U → I = [0, 1]\}\) is said to be a smooth partition of unity on open set \(U ⊂ ℝ^n\) if

  1. The supports \[\{\mathop{\mathrm{supp}}ρ: ρ ∈ \mathcal R\}\] is locally finite in \(X\);

  2. For all \(x ∈ U\), \[\sum_{ρ ∈ \mathcal R} ρ(x) = 1,\] where the notation is defined as \[\sum_{ρ ∈ \mathcal R}ρ(x) := \sum_{ρ∈\mathcal R, ρ(x) ≠ 0}ρ(x)\] which is a finite sum by above local finiteness.

Theorem 1.1 There exists a partition of unity for all open sets \(U \in \mathbb R^n\).