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 $\mathbf{Top}$.