Separate versus joint continuity: A tale of four topologies

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