Definition:Partition of Unity (Topology)/Subordinate

Definition
Let $X$ be a topological space.

Let $\AA = \set {\phi_\alpha : \alpha \in A}$ be a partition of unity.

Let $\BB = \set {U_\beta: \beta \in B}$ be an open cover of $X$.

Let the set $\set {\map {\operatorname {supp} } {\phi_\alpha}^\circ: \alpha \in A}$ of interiors of supports be a refinement of $\BB$.

Then $\AA$ is said to be subordinate to the cover $\BB$.