In Strøm's paper "The Homotopy Category is a Homotopy Category" he proves (Lemma 4) that if $Y$ is compact and if $i:A\to X$ is a cofibration, then the induced map $$i_*: A^Y \to X^Y$$ is also a cofibration.

The proof goes like this: he's already shown that $i:A\to X$ is a cofibration if and only if there is a function $u: X\to I$ such that $A = u^{-1}(0)$ and a deformation $H:X\times I \to X$ of $X$ that is constant on $A$ and pushes $U = u^{-1}([0,1))$ into $A$. Then he defines $$v(\alpha) = \sup \{ i\circ \alpha(Y) \} \qquad \mathrm{and}\qquad K (\alpha, t) (y) = H(\alpha(y), t).$$ This shows that $i_*$ is a cofibration since $v^{-1}(0) = A^Y$ and $K$ deforms $V = v^{-1}([0,1))$ into $A^Y$. When I try to prove that $v$ is continuous, it's very helpful to have the compactness of $Y$.

My question is: can the compactness hypothesis be dropped, perhaps if we work with compactly generated spaces?

EDIT: Ok, I see the problem with noncompact domains.
The space $I^{\mathbb{R}}$ is a good example. Functions in the set $\mathcal{U}(C,V)$ ($C\subseteq \mathbb{R}$ compact, $V\subseteq I$ open) are only limited on $C$, and so can take very large values elsewhere. You can cut this down a bit by taking intersections, and this does the job for compact domains, but you can only take finitely many. These spaces being as nice as can be, this is a dead question.

3 Got rid of the empty set in the name of Arne Strøm
2 Corrected Norwegian vowel

In Str$\varnothing$m's Strøm's paper "The Homotopy Category is a Homotopy Category" he proves (Lemma 4) that if $Y$ is compact and if $i:A\to X$ is a cofibration, then the induced map $$i_*: A^Y \to X^Y$$ is also a cofibration.

The proof goes like this: he's already shown that $i:A\to X$ is a cofibration if and only if there is a function $u: X\to I$ such that $A = u^{-1}(0)$ and a deformation $H:X\times I \to X$ of $X$ that is constant on $A$ and pushes $U = u^{-1}([0,1))$ into $A$. Then he defines $$v(\alpha) = \sup \{ i\circ \alpha(Y) \} \qquad \mathrm{and}\qquad K (\alpha, t) (y) = H(\alpha(y), t).$$ This shows that $i_*$ is a cofibration since $v^{-1}(0) = A^Y$ and $K$ deforms $V = v^{-1}([0,1))$ into $A^Y$. When I try to prove that $v$ is continuous, it's very helpful to have the compactness of $Y$.

My question is: can the compactness hypothesis be dropped, perhaps if we work with compactly generated spaces?

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