Continuity of separately continuous group actions in p-spaces

Abstract

Let ƒ:X × Y → Z be a separately continuous mapping, where X is a Baire p-space and Z a completely regular space, and let y ϵ Y be a q-point. We show that 1. (i) ƒis strongly quasicontinuous at each point of X × { y}, 2. (ii) if Z is a p-space, then ƒ is subcontinuous at each point of A × { y}, where A is a dense subset of X. Then, we use (i) and (ii) to prove that every separately continuous action of a left topological group, which is a Baire p-space, in a p-space, is a continuous action. In particular, every semitopological group, which is a Baire p-space, has a continuous multiplication.