Namioka property of generalized ordered spaces

Abstract

A topological space $X$ is called Namioka, if for every compact space $K$ and every separately continuous function $f:X\times K\to\mathbb R$ there exists a dense $G_\delta$-set $A\subseteq X$ such that $f$ is jointly continuous at every point of $A\times K$.

We introduce a notion of a topological space with countable resolvable set condition (i.e. in every separable subspace of such space each well-ordered strictly increasing (or decreasing) family of resolvable sets is at most countable) and prove that every hereditarily Baire perfect generalized ordered space with countable resolvable set condition is Namioka.