Exercise 19, § 1

Let B be a topological space, let $u: X\rightarrow \mathbf{I}$ be a continuous mapping and let $A =\overset{-1}{u}(0)$.

a) Exercise prove that the mapping $p$ of $X\times \mathbf{I}$ into itself given by $p(x, t) = (x, tu(x))$ is closed.

b) Let Y be a topological space and let $\sigma : X\times \mathbf{I}\rightarrow Y$ be a homotopy fixed on A. Let $\tau : X\times \mathbf{I}\rightarrow Y$ be the map given by $\tau (x, t) =\sigma (x$, inf(1$, t/u(x)$)) if $u(x)>0$ and $\tau (x, t) =\sigma (x,1)$ otherwise. Exercise prove that $\tau$ is continuous.

c) Let C be a subspace of X; suppose that the pair $(X,C)$ enjoys the homotopy extension property. Let $f: X\rightarrow Y$ be a continuous mapping and let $\sigma : C\times \mathbf{I}\rightarrow Y$ be a homotopy of origin $f|C$ which is fixed on $A\cap C$. Exercise prove that there exists a homotopy $\tau : X\times \mathbf{I}\rightarrow Y$ of origin $f$ which extends $\sigma$ and which is fixed on A.