Brown-Booth-Tillotson theory for classes of exponentiable spaces

Brown, Booth and Tillotson introduced the C-product, or the BBT C-product, for any class C of topological spaces. It is proved that any topological space is exponentiable with respect to the BBT C-product if and only if C is a subclass of the class of exponentiable spaces. The topology of the function space is induced by a canonical manner making use of the exponential topology for the spaces in C. It is not the C-open topology in general. The function space defined by this method enjoys good properties for algebraic topology. A necessary and sufficient condition on the class C is obtained for the exponential function to be a homeomorphism with the BBT C-product.