On the Bitopological Extension of the Bing Metrization Theorem

Abstract

Based on a Junnila's paracompactness characterization we give a definition of pairwise paracompact space which permits us to prove that a bitopological space is quasi-metrizable if, and only if, it is a pairwise developable and pairwise paracompact space. An easy consequence of this result is the biquasi-metric form of the Morita metrization theorem. We also give some results on open mappings and strong quasi-metrics.1980 Mathematics subject classification (Amer. Math. Soc): primary 54 E 35, 54 E 55; secondary 54 C 10, 54 D 18, 54 E 30.

Covering properties for LQ𝒰s with nested bases

“…Since a metrizable space is paracompact, it is a reasonable requirement for a quasi-metrizable space to be pairwise paracompact with respect to any definition of the pairwise paracompactness. Nevertheless, among the relative definitions in M. C. Datta [1], P. Fletcher [2], C. Konstadilaki-Savopoulou and I. L. Reilly [6], T. G. Raghavan [10], S. Romaguera and J. Marín [11] and M. Ganster and I. L. Reilly [4], only the last two satisfy this demand, although all of them coincide with the "paracompactness" in the case where the bitopological spaces are reduced to simple ones. Furthermore, J. Williams [12,Theorem 2.8] demonstrated that locally uniform spaces with nested bases are paracompact.…”

“…Furthermore, J. Williams [12,Theorem 2.8] demonstrated that locally uniform spaces with nested bases are paracompact. We show that, according to the definitions introduced in [4,11], the pairwise paracompactness is directly derived from quasi-uniformities with a nested base. We will symbolize in the text: [11]-or [4]-pairwise paracompactness, respectively.…”

“…The [11]-pairwise paracompactness. A pairwise regular space (X, ᏼ, ᏽ) is pairwise paracompact if and only if given a pair cover (Ᏻ, Ᏼ), there is for every x a sequence {U n [x] : n ∈ N} of ᏼ-neighborhoods and a sequence…”

“…We come now to discuss the [4,11]-pairwise paracompactness. As usual we denote by ᐁ * the uniformity which is the supremum of the quasi-uniformities ᐁ and ᐁ [x] for every n ∈ ω.…”

“…Stone's type theorem for the pairwise paracompactness works well for some definitions like, for instance, those introduced in [4,11] whilst it does not for some others, as in [1,2,6,10]. The local quasi-uniformity with a nested base which is constructed in Theorem 2.5 assures the pairwise paracompactness.…”