[Math] Partition of unity subordinate to an open cover

Question

Is a partition of unity $\{\phi_{\alpha}\}_{{\alpha}\in A}$ subordinate to an open cover $\mathcal{O}=\{O_{\alpha}\}_{{\alpha}\in A}$ of a smooth manifold $M$ also a cover of $M$? I ask because subordinate means $\operatorname{supp}(\phi_{\alpha})\subset O_{\alpha}$ for each $\alpha \in A$. If $\{\phi_{\alpha}\}_{{\alpha}\in A}$ is not a cover then how can a partition of uinity take local properties to global properties which is what they were constructed for? I question that $\{\phi_{\alpha}\}_{{\alpha}\in A}$ is a cover since $\operatorname{supp}(\phi_{\alpha})\subset O_{\alpha}$; i.e. the values of the function $\phi_{\alpha}$ are identically $0$ for a proper subset of $O_{\alpha}$

Answer 1

The collection of open sets $U_a=\{\phi_a>0\}$ does yield a cover of your manifold (which is perhaps the intuition you seek?).