Let $(X, A)$ be a cofibration, with $X$ compactly generated. This is equivalent to the fact that $A$ is a NDR of $X$, i.e., there exist two functions $\phi \colon X \rightarrow I$ e $H \colon X \times I \rightarrow X$ such that $A = \phi^{-1}(\{0\})$, $H(x, 0) = x$ and $H(a, t) = a$ for every $x \in X, a \in A, t \in I$, and $H(x, 1) \in A$ for every $x \in \phi^{-1}([0, 1))$. We call $U := \phi^{-1}([0, 1))$. The definition does not imply that $A$ is a (strong) deformation retract of $U$, because it is not necessary that $H(U \times I) \subset U$. Actually, in Strom's book ''Modern classical homotopy theory'', theorem 5.20 p. 104, it is stated that $A$ is a deformation retract of $U$, but I didn't manage to prove that the function $H$, defined in problem 5.21 p. 105, satisfies $H(U \times I) \subset U$. My question is the following.
Question: If $(X, A)$ is a NDR pair and $X$ is compactly generated, is it true that $H$ and $\phi$ can be chosen in such a way that $A$ is a (strong) deformation retract of U? (This implies that $(X, A)$ is a good pair in the sense of Hatcher's book.) If not, what is a counter-example? Moreover, if the answer is negative, is there a cofibration $(X, A)$ such that $A$ is not a (strong) deformation retract of any neighborhood, even different from $\phi^{-1}([0, 1))$?
Actually, in May's book ''The geometry of iterated loop spaces'', definition A.1 p. 85, an NDR-pair is called strong if $\phi(H(x, t)) < 1$ when $\phi(x) < 1$, which is equivalent to $H(U \times I) \subset U$. Therefore my question can be formulated in the following way: is every NDR-pair (at least when $X$ is compactly generated) strong? If not, what is a counter-example?
