Finite Partitions of Unity from Urysohn Functions

From Open Covers to Continuous Weights

A finite open cover records where local descriptions are available, but it does not itself provide continuous weights for combining those descriptions. A partition of unity supplies such weights: at each point the weights are nonnegative and sum to one, while each weight is confined to one member of the cover. Normality gives enough room to construct these functions by first shrinking the cover and then applying Urysohn’s lemma.

Throughout, a normal space \((X,\mathcal{T})\) means a \(T_1\) space in which any two disjoint closed sets have disjoint open neighborhoods. For an open cover \(U_1,\dots,U_n\), “finite” means \(n\geq 1\) is an integer and \(X=\bigcup_{i=1}^n U_i\). The support of a real-valued function \(f\) is the closed set \(\operatorname{supp}(f)=\overline{\{x\in X:f(x)\ne0\}}\).

The support condition is stronger than merely requiring \(\varphi_i(x)=0\) when \(x\notin U_i\): it requires the closure of the nonzero set to stay inside \(U_i\). The construction below achieves this stronger condition.

Shrinking a Finite Open Cover

We first record the consequence of normality that provides room around a closed set. If \(C\) is closed, \(U\) is open, and \(C\subseteq U\), then there is an open set \(W\) such that \(C\subseteq W\subseteq\overline{W}\subseteq U\). Indeed, separate the disjoint closed sets \(C\) and \(X\setminus U\) by disjoint open sets \(G,H\), with \(C\subseteq G\) and \(X\setminus U\subseteq H\). Since \(H\) is open and misses \(G\), it also misses \(\overline{G}\): each point of \(H\) has the neighborhood \(H\) disjoint from \(G\). Thus \(C\subseteq G\subseteq\overline{G}\subseteq X\setminus H\subseteq U\), so \(W=G\) works.

Proof. We use induction on \(n\). If \(n=1\), the cover condition says \(U_1=X\), and \(F_1=X\) works. Suppose \(n\geq2\) and the result holds for covers with \(n-1\) members. The set \(C=X\setminus\bigcup_{i=2}^n U_i\) is closed and is contained in \(U_1\). By the preceding consequence of normality, choose open \(W\) with \(C\subseteq W\subseteq\overline{W}\subseteq U_1\). The closed set \(Y=X\setminus W\), with its subspace topology, is normal: disjoint relatively closed subsets of \(Y\) are closed in \(X\) because \(Y\) is closed, and open neighborhoods separating them in \(X\) restrict to separating open neighborhoods in \(Y\). The sets \(U_i\cap Y\), for \(2\leq i\leq n\), form an open cover of \(Y\). By induction there are sets \(F_i\) closed in \(Y\), contained in \(U_i\cap Y\), and covering \(Y\). Since \(Y\) is closed in \(X\), each \(F_i\) is closed in \(X\). Set \(F_1=\overline{W}\), which is closed and contained in \(U_1\). The sets \(W\) and \(Y=X\setminus W\) cover \(X\); \(W\subseteq F_1\), and \(Y\) is covered by \(F_2,\dots,F_n\). Therefore \(F_1,\dots,F_n\) cover \(X\), completing the induction. \(\square\)

Apply the closed shrinking lemma, then apply the same neighborhood consequence to each pair \(F_i\subseteq U_i\). This gives open sets \(V_i\) with \[ F_i\subseteq V_i\subseteq\overline{V_i}\subseteq U_i. \] The closed sets \(F_i\) still cover \(X\). These intermediate open sets ensure that the eventual supports, not just the nonzero values, stay within the original cover.

The Partition-of-Unity Construction

We use Urysohn’s lemma in its standard form: if \(A\) and \(B\) are disjoint closed subsets of a normal space, there is a continuous \(f:X\to[0,1]\) with \(f=0\) on \(A\) and \(f=1\) on \(B\). For each \(i\), the sets \(X\setminus V_i\) and \(F_i\) are disjoint and closed. Choose a continuous \(f_i:X\to[0,1]\) equal to \(0\) on \(X\setminus V_i\) and \(1\) on \(F_i\). Define \[ S(x)=\sum_{j=1}^n f_j(x). \] Because the \(F_i\) cover \(X\), for each \(x\in X\) some \(F_i\) contains \(x\), so \(f_i(x)=1\) and \(S(x)\geq1\). The denominator is therefore positive everywhere.

Proof. Define \(\varphi_i(x)=f_i(x)/S(x)\). Each \(f_i\) and \(S\) is continuous, and \(S(x)\geq1\), so division gives a continuous function \(\varphi_i:X\to[0,1]\). The range is in \([0,1]\) because \(f_i(x)\geq0\) and \(S(x)\geq f_i(x)\). At every \(x\), \[ \sum_{i=1}^n\varphi_i(x)=\sum_{i=1}^n\frac{f_i(x)}{S(x)} =\frac{S(x)}{S(x)}=1. \] If \(\varphi_i(x)\ne0\), then \(f_i(x)\ne0\), so \(x\in V_i\), since \(f_i\) vanishes on \(X\setminus V_i\). Consequently \(\{x:\varphi_i(x)\ne0\}\subseteq V_i\), and taking closures gives \(\operatorname{supp}(\varphi_i)\subseteq\overline{V_i}\subseteq U_i\). These are exactly the required conditions. If \(X=\varnothing\), the unique functions with domain \(X\) satisfy the conditions vacuously. \(\square\)

Worked Constructions

Why Normality Matters and a Common Pitfall

The construction separates two tasks. The finite closed shrinking ensures that every point lies in a closed set \(F_i\) assigned to some \(U_i\). Urysohn’s lemma then produces a function that is \(1\) on that closed set and zero away from the larger neighborhood \(V_i\). The closed cover is what prevents the normalizing sum from vanishing; the extra containment \(\overline{V_i}\subseteq U_i\) is what controls the supports.

Common pitfall: assuming the initial functions already sum to one. In the interval example, at \(x=\frac12\) the raw values are \(h_1=h_2=\frac16\), so their sum is \(\frac13\), not \(1\). Treating \(h_1,h_2\) as a partition without normalization gives the wrong total. The correction is to divide both by the same positive sum: \(\varphi_1=(\frac16)/(\frac13)=\frac12\) and \(\varphi_2=(\frac16)/(\frac13)=\frac12\). Positivity of that denominator must be proved at every point before this division is justified.

Once constructed, the functions can be used as continuous weights: if local continuous quantities are available on the corresponding cover members, the functions indicate how much each member contributes at a point. The finite sum-to-one condition preserves the total weight, while the support condition prevents a function from contributing outside its assigned region. No compactness assumption is needed for this finite-cover construction.