TITLE: The union of all coreflective Cartesian closed subcategories of $\mathbf{Top}$
QUESTION [7 upvotes]: Very often, in topology, one restricts to a coreflective Cartesian closed subcategory of $\mathbf{Top}$ in order to freely use exponential laws for mapping spaces, which imply things like "the internal categorical product of quotient maps is a quotient map." I have a situation where an argument works regardless of the particular "convenient" subcategory that I pick. Hence, as long as a space lies in some coreflective Cartesian closed subcategory, I may apply my argument.
What is the class of topological spaces, that lie in some coreflective Cartesian closed subcategory of $\mathbf{Top}$?
Update: It is well-known that every class of spaces generates a Cartesian closed subcategory of $\mathbf{Top}$ (See Booth-Tillotson) but it may not be coreflective. On the other hand, many coreflective categories like the category of locally path-connected spaces are not Cartesian closed. As pointed out by David White below, it is a result of Juraj Cincura that there is no largest coreflective Cartesian closed subcategory of $\mathbf{Top}$. 
I'd be content to just to know that there is a space that does not lie in any coreflective Cartesian closed subcategory of $\mathbf{Top}$.

REPLY [5 votes]: I contacted Juraj Činčura and he kindly wrote back and directed me to the following observation that is a consequence of results in the paper that David White noted in his answer.
Cartesian closed coreflective subcategories of the category of topological spaces, Topology and its Applications 41 (1991) 205-212. 
For an infinite cardinal $a$, let $C(a)$ be the space defined on set $a\cup \{a\}$ where $V\subseteq C(a)$ is open if and only if $V\subseteq a$ or if $a\in V$ and $|a\backslash V|<a$.
Činčura's Proposition 2.1 states that the coreflective hull of $C(a)$ in $\mathbf{Top}$ or $\mathbf{Haus}$ is Cartesian closed. However, Proposition 3.1 states that if $a$ is a strictly larger cardinal than $b$ and class $\mathscr{B}$ contains both $C(a)$ and $C(b)$, then the coreflective hull of $\mathscr{B}$ in $\mathbf{Top}$ or $\mathbf{Haus}$ is not Cartesian closed. Since $C(a)$ and $C(b)$ are both quotients of $C(a)\times C(b)$, this implies that if $a>b$, then any coreflective subcategory in $\mathbf{Top}$ or $\mathbf{Haus}$ containing the space $C(a)\times C(b)$ is not Cartesian closed.
Hence, there is no coreflective Cartesian closed subcategory of $\mathbf{Top}$ or $\mathbf{Haus}$ containing $C(a)\times C(b)$ if $a\neq  b$.
This is a specific example showing that there is no largest coreflective Cartesian subcategory of $\mathbf{Top}$ or $\mathbf{Haus}$.