During some self-study, I came across the following problem in Spanier's Algebraic Topology:
Statement: Suppose $X$ is a normal space, and $A$ is a closed subspace of $X$. Let $f\colon X \to Y$ be a continuous map (where $Y$ is an arbitrary space). Suppose there is a homotopy $G \colon A \times I \to Y$ with $G(x, 0) = f(x)$ for all $x \in A$. Suppose further that $G$ can be exentended to a homotopy $G' \colon U \times I \to Y$ with $G'(x, 0) = f(x)$ for all $x \in U$, where $U$ is an open neighborhood of $A$ in $X$. Then $G$ can be extended to a homotopy $H \colon X\times I \to Y$ with $H(x, 0) = f(x)$ for all $x \in X$.
Attempted Solution 1: My thought process is a bit ad-hoc on this problem (I don't have a very strong background in point-set topology). I'm trying to work backwards. Since $X$ is specified to be normal, I'm thinking I should find disjoint closed subsets, separate them by neighborhoods, and then somehow use this separation to construct my homotopy on $X$. Clearly, the complement $U^c$ of $U$ is closed and disjoint from $A$. Let $U_1$ and $U_2$ be disjoint neighborhoods of $U^c$ and $A$, respectively.
Now I get a bit stuck on how to use $U_1$ and $U_2$ to define the desired homotopy. My intuitive idea is to have $H|_{U_1 \times I}$ be the constant homotopy at $f$, and $H|_{U_2 \times I} = G'|_{U_2\times I}$. How can I define $H$ on $(U_1 \cup U_2)^c$ so that $H$ is continuous? Is this idea doomed to fail?
Attempted Solution 2: I read on Wikipedia that if $X$ is normal, then any locally finite open cover $\{U_i\}_{i \in I}$ has a subordinate partition of unity $\{\rho_i\}_{i \in I}$. I thought I could use this as follows: $\{U, A^c\}$ is a (locally) finite open cover, so I have some subordinate partition of unity $\{\rho_U, \rho_A\}$. Then I was thinking I could use this partition of unity to define $H$ by "tapering" $G'$, so that I end up at the constant homotopy on $f$. Please excuse my imprecise language here, I don't think I have the necessary vocabulary to express what my intuition is telling me. I get stuck, however, because I don't seem to have control on the support $\rho_U$ and $\rho_A$.
Any suggestions would be greatly appreciated. Thank you in advance.
This is a consequence of Tietze Uryshon.
Let $F$ the complement of $U$ it is closed, there exists a real function $h$ whose restriction to $F$ is $0$ and whose restriction to $A$ is $1$,
set $H(x,t)=G'(x,h(x)t), x\in U$, $H(x,t)=f(x), x\in F$.