Locally finite partitions of unity and subordination to an open cover

Let $X$ be a topological space and let $\mathcal U$ be an open cover of $X$. A family $\{\varphi_s:X\to[0,1]\}_{s\in S}$ is a partition of unity when each $\varphi_s$ is continuous, the family of cozero sets $\{\operatorname{coz}(\varphi_s)\}_{s\in S}$ is locally finite, and $\sum_{s\in S}\varphi_s(x)=1\quad\text{for every }x\in X.$ The sum is unambiguous because local finiteness says that only finitely many summands are nonzero near, and hence at, any fixed point.

It is subordinate to $\mathcal U$ when for every $s\in S$ some $U\in\mathcal U$ contains the support $\operatorname{supp}(\varphi_s):=\overline{\operatorname{coz}(\varphi_s)}.$ Here cozero sets and zero sets have the meanings of Zero sets and cozero sets of continuous real-valued functions.

The finite case is included: if $S$ is finite, the cozero family is locally finite automatically. The definition does not require $X$ to be Hausdorff; Hausdorffness enters the existence theorem through shrinking and Urysohn's lemma.