Separate versus joint continuity: A tale of four topologies

Several naturally occurring topologies on the product X x Y of the Tychonoff spaces X and Y are studied; each is stronger than the product topology tau. These include the cross topology gamma consisting of sets meeting each horizontal and vertical fiber in a set open in the subspace topology induced by tau; the weak topology a determined by the separately continuous real-valued functions with domain X x Y; and the weak topology determined by certain special separately continuous functions. Functorial relations between gamma and a are described. Sufficient conditions for separately continuous functions to be jointly continuous on a dense subspace of (X x Y,tau) are given. The topological structure of (X x Y,sigma) is studied in detail. (C) 1999 Elsevier Science B.V. All rights reserved.