Dependence on ℵ coordinates of separately continuous functions of many variables and its analogs

Abstract

We introduce some cardinal functions on the product X1×⋯×Xn of topological spaces Xk, which are formulated in terms of the type of local finiteness of families of open sets. Using these cardinal functions, we obtain necessary and sufficient conditions that every separately continuous function or strongly separately continuous function f:X1×⋯×Xn→R depends on ℵ coordinates, where every space Xk is a strongly countably Čech complete space.