Exponential homeomorphisms in the category of topological spaces with base point

Topologies on base point preserving function spaces are studied making use of the Brown–Booth–Tillotson C-smash product and the topologies on function spaces defined by the classes C of exponentiable spaces. Some conditions on the classes C are obtained for exponential bijections and exponential homeomorphisms in the category of topological spaces with base point. Conditions for exponential homeomorphisms are obtained for CW-complexes whose cellular structure implies precise results.