Showing a subcategory of $\mathbf{Top}$ is Cartesian-Closed We start with some preliminary definitions (necessary because there is not much literature on this): 


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*a test map is a continuous function $\varphi:V\rightarrow X$ where $V$ is an open subspace of $\mathbb{R}^n$ for some $n$.

*Given a topological space $X$, we say that a subset $U$ of $X$ is numerically open if $\varphi^{-1}(U)$ is open for every test map $\varphi$.

*A topological space $X$ is said to be numerically generated if a subset $U$ of $X$ is open if and only if it is numerically open.

*Given any topological space $X$, we define the numericalization of $X$, denoted $X^\#$, to be the set underlying $X$ equipped with the final topology induced by the family of test maps into $X$.

*The numerically generated product of two numerically generated spaces $X$ and $Y$ is the numericalization of the product space $X\times Y$ (henceforth simply denoted $X\times Y$).

*The numerically generated mapping space of the numerically generated spaces $X$ and $Y$, denoted $Map(X,Y)$, is the collection of all continuous maps from $X$ to $Y$ equipped with the numericalization of the compact-open topology.

*The category $\mathbf{Num}$ is the full subcategory of $\mathbf{Top}$ with objects as numerically generated spaces; this has as coproduct the disjoint union with the disjoint union topology (it is not difficult to show that the disjoint union of numerically generated spaces is numerically generated) and product as the numerically generated product.


I wish to show that $\mathbf{Num}$ is a cartesian-closed category.  Ultimately, this simply boils down to showing that the numerically generated mapping space is an exponential object in $\mathbf{Num}$.
Naturally, I would want to use the typical bijection $h: Hom(X\times Y,Z)\rightarrow Hom(X,Z^Y)$ given by currying, but it remains to show that given a continuous map $f:X\times Y\rightarrow Z$, that the map $g:X\rightarrow Map(Y,Z)$ defined by $g(x)=f(x,-)$ is continuous (as well as that $f(x,-)$ is continuous too).  However, I'm tripping up on showing this.
 A: I know this answer is coming with a few years of delay, but there is a paper by  Escardó, Lawson and Simpson called comparing Cartesian closed categories of (core)compactly generated spaces that gives a very general result.
I'll highlight the main parts here for your convenience.
Let $\mathbf{Top}$ be the category of topological spaces and let $I$ be a small full subcategory of $\mathbf{Top}$.
Definition: A topological space $X$ is said to be $I$-generated, if it is a colimit of objects in $I$. Equivalently, $X$ is $I$-generated if it has the final topology with respect to the family of all maps $f\colon C\to X$ from objects in $I$ to $X$.
Proposition:
Assume that all the objects in $I$ are core-compact (aka exponentiable in $\mathbf{Top}$).
Moreover, assume that finite products of objects in $I$ are $I$-generated.
Then, the category of $I$-generated spaces is Cartesian closed.
In particular, numerically generated spaces are Cartesian closed.
The way I like to see this is that numerically generated spaces coincide with $\Delta$-generated spaces, where we take $\Delta$ to be the full subcategory of $\mathbf{Top}$ spanned by the geometric realisations of the standard simplices.
Then, since realisation commutes with products of standard simplices, the claim follows.
