Definition:Partition of Unity (Topology)/Subordinate

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Definition

Let $X$ be a topological space.

Let $\left\{{\phi_\alpha : \alpha \in A}\right\}$ be a partition of unity.

Let $\mathcal B = \left\{{U_\beta: \beta \in B}\right\}$ be an open cover of $X$.

Suppose the set $\left\{{\operatorname{supp} \left({\phi_\alpha}\right)^\circ : \alpha \in A}\right\}$ of interiors of supports is a refinement of $\mathcal B$


Then $\mathcal A$ is said to be subordinate to the cover $\mathcal B$.