Strict complete regularity in the categories of bitopological spaces and ordered topological spaces

Abstract

We introduce the notion of a strictly completely regular bitopological space and show that the category of strictly completely regular bitopological spaces is isomorphic to the category of strictly completely regular ordered spaces. It is shown by means of examples that the category of strictly completely regular bitopological spaces is more flexible than its counterpart in ordered spaces. We also determine quasiuniformities that induce strictly completely regular bitopologies.

A note on the Universal bicompactification of strictly completely regular bitopological spaces

“…Consider the bitopological space R b = (R, u, l). We know that R b is a strictly completely regular bitopological space [4]. By Theorem 1, we have that…”

“…The following strengthening of complete regularity in bitopological spaces was given in [4]. Definition 1.…”

“…In [4] it was shown that the functors I and S coincide on strictly completely regular ordered spaces.…”

Ordered separation axioms and the Wallman ordered compactification

“…A bitopological space is pairwise completely regular (see [9]) if and only if it admits a compatible quasi-uniformity U in the sense that τ (U) is the first topology and τ (U −1 ) is the second topology. An ordered topological space is completely regularly ordered (see [7], [16]) if and only if there is a quasi-uniformity U with U =≤ and τ (U) ∨ τ (U −1 ) = τ . Such spaces are always convex and T 1ordered.…”