Confusion about topology on space of maps In this set of notes on compactly generated spaces by Charles Rezk, the author states the following result:
Proposition 8.6. If $X$ and $Y$ are k-spaces and $Z$ is any space, then a function $f: X \times^k Y \to Z$ is continuous if and only if its adjoint $\widetilde{f}: X \to \mathrm{Map}(Y,Z)$ is defined and continuous.
Here, $X \times^k Y$ denotes the compactly generated product, that is, the $k$-ification of the usual cartesian product of topological spaces, and the topology on the set of maps $\mathrm{Map}(Y,Z)$ is defined as follows: a subset $U \subseteq \mathrm{Map}(Y,Z)$ is open if and only if, for every compact Hausdorff space $K$ and every map $t: K \times Y \to Z$, the set $\widetilde{t}(U)$ is open in $K$, where $\widetilde{t}: K \to \mathrm{Map}(Y,Z)$ is the adjoint function of $t$.
I have a question about the forward direction of the statement.
The proof is as follows:

*

*The adjoint $\widetilde{f}$ really takes values in $\mathrm{Map}(Y,Z)$, because $\widetilde{f}(x): Y \to Z$ can be written as the composition $f \circ i_x$, where $i_x: Y \to X \times^k Y$ is the map defined as $i_x(y)= (x,y)$ for every $y \in Y$. Notice that $i_x$ is continuous when considering $X \times^k Y$ as the codomain because it can be regarded as the map induced by the constant map $Y \to X$ sending every point to $x$ and the identity $\mathrm{id}_Y: Y \to Y$.
Here we are using the fact that $\times^k$ is the categorical product in the category $\mathsf{CG}$ of compactly generated spaces and continuous maps, so it is clear that both $X$ and $Y$ need to be compactly generated.


*If $U \subseteq \mathrm{Map}(Y,Z)$ is open, and $t: K \to X$ is a map, with $K$ compact and Hausdorff, then we have the equality $t^{-1}(\widetilde{f}^{-1}(U)) = \widetilde{g}^{-1}(U)$, where $g = f \circ (t \times \mathrm{id}_Y)$.
This is essentially the naturality of the exponential adjunction.


*$t \times \mathrm{id}_Y$ can be seen as a map of type $K \times Y \to X \times^k Y$ because the usual cartesian product $K \times Y$ is already compactly generated.
This means that $g$ is a map of type $K \times Y \to Z$, so $\widetilde{g}^{-1}(U)$ is open in $K$ by the definition of the topology on $\mathrm{Map}(Y,Z)$.


*This shows that $\widetilde{f}^{-1}(U)$ is compactly open in $X$, but since $X$ is compactly generated by  hypothesis, it follows that $\widetilde{f}^{-1}(U)$ is open.
My question is: doesn't a similar argument work if we suppose only $X$ is compactly generated, but suppose instead that $f$ is of type $X \times Y \to Z$? Under these hypothesis, the "tricky" parts in the proof above become automatic: $\widetilde{f}$ takes values in $\mathrm{Map}(Y,Z)$ by standard properties of the cartesian product, and $t \times \mathrm{id}_Y$ is a map of type $K \times Y \to X \times Y$ automatically, no need to mess with the compactly generated product $X \times^k Y$.
Is this correct? I think it is, but since this is the only reference I found where a topology is defined directly on $\mathrm{Map}(Y,Z)$ instead of $k$-ifying the test-open topology or the space of $k$-continuous maps, I'm a little uncertain.
Moreover, this alternative result implies another one which feels wrong to me.
Corollary. For any spaces $X$ and $Y$, the space $\mathrm{Map}(Y,Z)$ is compactly generated.
Proof. Let $A \subseteq \mathrm{Map}(Y,Z)$ be a compactly open subset.
If $K$ is compact Hausdorff and $t: K \times X \to Y$ is a map, since every compact Hausdorff space is compactly generated, by the alternative result above the adjoint $\widetilde{t}: K \to \mathrm{Map}(X,Y)$ is continuous, so $\widetilde{t}^{-1}(A)$ is open in $K$, which means that $A$ is open in $\mathrm{Map}(X,Y)$.
 A: Your proofs are correct.
Regarding the corollary that seems wrong to you, it is actually correct. The intuitions is that enlarging a generating family of functions inducing a topology by continuous functions for that topology results in a generating family of continuous functions. In particular, the topology on $\mathrm{Map}(Y,Z)$ is induced by functions out of compact Hausdorff sets, hence is generated by all continous functions out of compact Hausdorff sets.
Regarding the forward direction of the proposition what you've proven is this. Suppose $\mathcal C$ is a collection of topological spaces, and $\mathrm{Map}_{\mathcal C}(Y,Z)$ is the set of continuous functions $Y\to Z$ with a topology such that for each $K\in\mathcal C$ we have $K\to\mathrm{Map}_{\mathcal C}(Y,Z)$ if $K\times Y\to Z$ is continuous. Then for $X$ with the property that $X\to\mathrm{Map}_{\mathcal C}(Y,Z)$ is continuous whenever $K\to X\to\mathrm{Map}_{\mathcal C}(Y,Z)$ is continous for each continuous $K\to X$ with $K\in\mathcal C$ (e.g. $\mathcal C$-generated $X$), we have $X\to\mathrm{Map}_{\mathcal C}(Y,Z)$ continuous if the corresponding $X\times Y\to Z$ is continuous.
This is immediate because $K\to X\to\mathrm{Maps}(Y,Z)$ correspond to $K\times Y\to X\times Y\to Z$ with $K\times Y\to X\times Y$ continous if $K\to X$ are continuous. Indeed, then $X\times Y\to Z$ continuous implies $K\times Y\to X\times Y\to Z$ continous, hence $K\to X\to\mathrm{Map}_{\mathcal C}(Y,Z)$ continous, hence $X\to\mathrm{Map}_{\mathcal C}(Y,Z)$ continuous.
The forward direction of the proposition is then a corollary because $X\times^{\mathcal C} Y$ is $X\times Y$ with the coarsest topology for which $K\to X\times Y$ for each $K\in\mathcal C$ are continuous. Consequently, there is a continuous map $X\times Y\to X\times^{\mathcal C}Y$ with the property that continuous maps $X\times^{\mathcal C} Y\to Z$ determine continuous maps $X\times Y\to Z$ such that both correspond to the same function $X\to\mathrm{Map}_{\mathcal C}(Y,Z)$. Consequently, $X\times^{\mathcal C} Y\to Z$ continuous for $\mathcal C$-generated $X$ implies $X\times Y\to Z$ is continuous, hence that the corresponding $X\to\mathrm{Map}_{\mathcal C}(Y,Z)$ is continuous.
A good reference for these topics is Escardó, Martín Hötzel et al. “Comparing Cartesian closed categories of (core) compactly generated spaces.” Topology and its Applications 143 (2004): 105-145.
