[1]:https://ncatlab.org/nlab/show/convenient+category+of+topological+spaces

You should look at this [ncatlab][1] exposition on "convenient categories of topological spaces", and references. One problem is still the difficulty of getting a **locally cartesian closed**  convenient category; Spanier's "quasitopologies" seemed a possibility but have  not been taken up, partly because the quasicategories on the 2 point set formed a class (comment from E. Dyer).