Numerable principal bundle

A topological principal bundle $P\to X$ is numerable if it has an open trivializing cover $\{U_i\}_{i\in I}$ and continuous functions $\phi_i:X\to[0,1]$ such that:

1. the supports $\operatorname{supp}\phi_i=\overline{\{x:\phi_i(x)\ne0\}}$ form a locally finite family;
2. $\sum_i\phi_i(x)=1$ for all $x\in X$;
3. $\{x:\phi_i(x)>0\}\subseteq U_i$ for each $i$.

The sum is locally finite, so it defines a continuous function. Numerability is the existence of these data; a particular cover and partition are not part of the bundle's structure.