2026/01/07 by Xiongping Dai, Congying Lv, Yuxuan Xie · 1 voice
Mathematics · #math.GN #math.DS
arxiv published 2026/01/07 · arxiv updated 2026/05/20
A space X is called a generalized Namioka space (gN-space), if for every compact space Y and every separately continuous function f\colon X× Y→ℝ, there exists at least one point x∈ X such that f is jointly continuous at each point of \x\× Y. We principally prove the following results: (1) If X=∏α∈ AXα is non-meager such that each factor is a separable space or each factor is a pseudo-metric space, then X is a gN-space. (2) If X is a separable space and Y a pseudo-metric space such that X× Y is Baire (resp. non-meager), then X× Y is an N-space (resp. a gN-space). (3) If X=∏α∈ AXα such that each factor is separable and ∏α∈ A^′Xα is a non-meager space for each countable subset A^′ of A, then X is a non-meager gN-space. (4) If X=∏α∈ AXα such that each factor has a countable π-base, then each tail set having the property of Baire in X is either meager or residual. If G is a gN right-topological group and X a locally compact regular space, or, if G is a separable first countable non-meager right-topological group and X× X a countably compact completely regular space, then any separately continuous action G\curvearrowright X is jointly continuous.