A Boolean-valued space approach to separation axioms and sobriety of bitopological spaces

TL;DR

This paper explores separation axioms and sobriety in bitopological spaces through Boolean-valued fuzzy topology, establishing new connections and a Hofmann-Mislove theorem for these spaces.

Contribution

It introduces a Boolean-valued approach to separation axioms and sobriety in bitopological spaces, linking them with fuzzy topology and establishing a Hofmann-Mislove theorem.

Findings

Proposed a new system of separation axioms using Boolean-valued specialization order.

Studied the relationship between d-sobriety and fuzzy topological sobriety.

Established a Hofmann-Mislove theorem for bitopological spaces.

Abstract

This paper presents a study of separation axioms and sobriety of bitopological spaces from the point of view of fuzzy topology via identifying bitopological spaces with topological spaces valued in the Boolean algebra of four elements. A system of separation axioms is proposed making use of Boolean-valued specialization order of bitopological spaces; The relationship between d-sobriety of bitopological spaces proposed by Jung and Moshier and sobriety of fuzzy topological spaces is studied; A Hofmann-Mislove theorem for bitopological spaces is established.