Paracompact spaces

Definition 9.1.2. A topological space \(X\) is called paracompact if for every open cover \(\{U_{i}\}_{i \in I}\) of \(X\) there exists a partition of unity subordinate to this cover, i.e. a collection of continuous maps \(\rho _{\alpha }\colon X \to [0,1]\) satisfying the following three conditions:

(1)

The family \((\rho _\alpha )_{\alpha }\) is locally finite: every point of \(X\) has a neighborhood on which all but finitely many \(\rho _\alpha \) vanish;

(2)

The resulting sum satisfies \(\sum _{\alpha } \rho _{\alpha }(x) = 1\) for all \(x\);

(3)

For each \(\alpha \), the support \(\supp (\rho _\alpha ):=\overline {\rho _{\alpha }^{-1}((0,1])}\) is contained in one of the open sets \(U_i\).