Exercise 31, § 1

Let B be a topological space. Let A and V be subspaces of B; one says that V is a halo of A if there exists a continuous mapping $\varphi : B\rightarrow \mathbf{I}$ which is equal to 1 at every point of A and to 0 at every point of $\complement V$. One says that a B-space $(E, p)$ has the property of extension of sections if, for every subspace A of B, every halo V of A and every section $s$ of $E_V$, there exists a section $s'$ of E such that $s'|A =s|A$. a) Let $(E, p)$ and $(E', p')$ be B-spaces; suppose that there exist morphisms of B-spaces, $f: E\rightarrow E'$ and $g: E'\rightarrow$ E, and a homotopy $\sigma : E\times \mathbf{I}\rightarrow E$ of origin Id$_E$ and of term $g\circ f$ such that $p(\sigma (x, t)) =p(x)$ for all $(x, t)\in E\times \mathbf{I}$. If the B-space $E'$ has the property of extension of sections, the B-space E has it also.

b) Let E be a B-space and let A be a subspace of B which is a retract of B. Suppose that the induced A-space $E_A$ has the property of extension of sections; prove that the B-space E has it also.

c) Let E be a B-space which has the property of extension of sections. Let

$f: B\rightarrow \mathbf{I}$ be a continuous mapping and let $V =\overset{-1}{f}(]0,1])$. Prove that the V-space $E_V$ has the property of extension of sections. (Let A be a subspace of V and let W be a halo of A in V; construct sequences $(A_n)$ and $(V_n)$ of subspaces of B satisfying the following properties: for every $n,A_n$ is contained in W, $V_n$ is a halo of $A_n$, and the intersection of the $A_n$ is equal to A.)