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 $\{{\phi }_s:X\to [0,1]{\}}_{s\in S}$ is a partition of unity when each ${\phi }_s$ is continuous, the family of cozero sets $\{\mathrm{coz}({\phi }_s){\}}_{s\in S}$ is locally finite, and ${\sum }_{s\in S}{\phi }_s(x)=1\text{for every }x\in X\mathrm{.}$ 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 $\mathrm{supp}({\phi }_s):=\overline{\mathrm{coz}({\phi }_s)}\mathrm{.}$ 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.