Exercise 32, § 1

Soit B un topological space and let $(E, p)$ be a B-space. Let $(V_i)_{i\in I}$ be a numerical covering of B such that, for every $i\in I$, the $V_i$-space $E_{V_i}$ possesses the property of extension of sections.

a) Suppose that there exists a continuous partition of unity $(f_i)$ on B, locally finite, such that $V_i=\overset{-1}{f_{i}}(]0,1])$ for every $i$. Let $g: B\rightarrow [0,1]$ be a continuous mapping. Put $A =\overset{-1}{g}(1)$ and $W =\overset{-1}{g}(]0,1])$ ; let $s$ be a section of E over W. Prove that there exists a section of E which coincides with $s$ on A. (Consider a maximal element of the set of pairs $(J, t)$, where J is a subset of I and $t$ is a section of E over $W\cup \bigcup_{i\in J}V_i$ which coincides with $s$ on A, endowed with the order relation given by $(J, t)\prec (J', t')$ if $J\subset J'$ and $t'$ extends $t.$)

b) Prove that the B-space E possesses the property of extension of sections.