Very often, in topology, one restricts to a [coreflective](https://ncatlab.org/nlab/show/coreflective+subcategory) [Cartesian closed](https://ncatlab.org/nlab/show/cartesian+closed+category) 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](https://projecteuclid.org/euclid.pjm/1102779712)) 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 $\mathbf{Top}$. 



The union of all coreflective Cartesian closed subcategories of $\mathbf{Top}$
The relevant paper is [Cartesian closed coreflective subcategories of the category of topological spaces](https://www.sciencedirect.com/science/article/pii/0166864191900046) by Juraj Cincura. The first line of the abstract says "Answering the first part of Problem 7 in [10] we prove that there is no largest Cartesian closed coreflective subcategory of the category Top of topological spaces and the same result for the category Haus of Hausdorff spaces."


Another great paper, by Vogt, is [Convenient Categories of Topological Spaces
for Homotopy Theory](https://link.springer.com/content/pdf/10.1007%2FBF01222616.pdf). It gives several examples, and argues the pros/cons of each. The author suggests that the coreflective subcategory generated by the locally compact spaces is large enough to contain everything he is interested in.


