Pairwise $k$-Semi-Stratifiable Bispaces and Topological Ordered Spaces

Abstract

In this paper, we continue to study pairwise ($k$-semi-)stratifiable bitopological spaces. Some new characterizations of pairwise $k$-semi-stratifiable bitopological spaces are provided. Relationships between pairwise stratifiable and pairwise $k$-semi-stratifiable bitopological spaces are further investigated, and an open question recently posed by Li and Lin in \cite{LL} is completely solved. We also study the quasi-pseudo-metrizability of a topological ordered space $(X,\tau ,\preccurlyeq )$. It is shown that if $(X,\tau ,\preccurlyeq )$ is a ball transitive topological ordered $C$- and $I$-space such that $\tau$ is metrizable, then its associated bitopological space $(X,{\tau }^{♭},{\tau }^{♮})$ is quasi-pseudo-metrizable. This result provides a partial affirmative answer to a problem in \cite{KM}.

13 pages