Function Spaces and Product Topologies

Introduction IN a previous paper (4) I defined ten product topologies o n l x T . In this paper five of these products are applied to problems on function spaces. All spaces will be Hausdorff spaces. The exponential law for function spaces with the compact-open topology is discussed in § 1. The main result (Theorem 1.6) is that the spaces X and (X) are homeomorphic for all X, Y, Z (in this paper ZxY will denote the product ZxsY defined in (4), and ZxY will denote the usual, cartesian, product). Hence the exponential law holds for ZxY if Z x 7 = ZxF , and this contains and explains many known results. We deduce also some new results. For example we prove that the answer is 'no' to Dr. S. Wylie's question: are the spaces (X) and (X) naturally homeomorphic? In § 2 we discuss the law (XxY) = XxY. This fails in general for products other than the cartesian. .§ 3 is the most important section. It advertises the category of Hausdorff spaces and functions continuous on compact subspaces (here called k-continuous functions). In § 4 the exponential law of § 3 is generalized to the category of .if-ads. I am indebted to a referee, whose comments stimulated a complete revision and extension of the original draft, and to Dr. M. G. Barratt for the inspiration of his conversation and example. I am also indebted to Dr. W. F. Newns and Professor A. Dold for helpful conversations.