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