Exercise 29, § 1

Let $B$ be a topological space. A covering $(V_i)_{i\in I}$ of B is said to be numerical if there exists a continuous partition of unity $(g_j)_{j\in J}$ on B, locally finite, such that for every $j\in J,\overset{-1}{g_{j}}(]0,1])$ is contained in one of the $V_i$.

a) Prove that B is paracompact if and only if every open covering of B is numerical.

b) Prove that B is normal if and only if every locally finite open covering of B is numerical.

c) Let $(V_i)$ be a numerical covering of B, let $B'$ be a topological space and let $f: B'\rightarrow B$ be a continuous mapping. Prove that the covering $(\overset{-1}{f}(V_i))$ of $B'$ is numerical.