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.