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

## TL;DR

This paper advances the understanding of pairwise $k$-semi-stratifiable bitopological spaces by providing new characterizations, exploring their relationships, and addressing open questions, while also examining quasi-pseudo-metrizability in topological ordered spaces.

## Contribution

It offers new characterizations of pairwise $k$-semi-stratifiable spaces, resolves an open question, and links topological ordered spaces with bitopological quasi-pseudo-metrizability.

## Key findings

- New characterizations of pairwise $k$-semi-stratifiable spaces.
- Complete resolution of an open question by Li and Lin.
- Establishment of quasi-pseudo-metrizability for certain 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 $(X,\tau^{\flat},\tau^{\natural})$ is quasi-pseudo-metrizable. This result provides a partial affirmative answer to a problem in \cite{KM}.