On the bicompletion of a partial quasi-metric space and $$T_{0}$$ T 0 -quasi-metric spaces
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On the bicompletion of a partial quasi-metric space and T0 -quasi-metric spaces Seithuti Moshokoa1
· Fanyana Ncongwane1
Received: 6 November 2018 / Accepted: 1 September 2020 © African Mathematical Union and Springer-Verlag GmbH Deutschland, ein Teil von Springer Nature 2020
Abstract The purpose of the paper is to extend the completion theory of a partial metric space to the context of an asymmetric setup, namely, a partial quasi-metric space. We present two bicompletions of a partial quasi-metric space by appealing to the associated partial metric space on one hand and while on the other hand an associated T0 -quasi-metric space is utilized. The two bicompletions are not necessarily the same but they coincide once the class of partial quasi-metric spaces considered is restricted to the class of T0 -quasi-metric spaces. Keywords Partial quasi-metric space · Bicompletion · T0 -quasi metric space · Quasi-uniform space Mathematics Subject Classification Primary 54E; Secondary 54E99 · 54E50 · 54E99
1 Introduction In functional analysis or real analysis the problem of completing a metric space, a uniform space and a normed linear space plays a crucial role. This problem has been extended to asymmetric mathematical structures, namely, quasi metric spaces [17], quasi-uniform spaces [4,12] and asymmetric normed linear spaces [5]. Recently, a similar problem is receiving much attention in partial metric spaces [3,6]. The initial work on the completion of a partial metric space appears in the paper [15], and more recent work on the completion of a partial metric space appears on the papers [3,6]. The purpose of this article is to extend the completion theory of partial metric spaces to partial quasi-metric spaces also referred to as quasi partial metric spaces [13]. These structures were introduced by Kunzi et al [10] where their applications to Groups and BCK-algebras was also presented. The completion of these structures was presented in the same article. Since then many authors studied more properties of
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Seithuti Moshokoa [email protected] Fanyana Ncongwane [email protected]
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Department of Mathematics and Statistics, Tshwane University of Technology, Pretoria 0008, South Africa
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S. Moshokoa, F. Ncongwane
partial quasi-metric spaces, with special emphasis on the extensions of the Banach contraction theorem to the context of partial quasi-metric spaces and its generalizations [1,8,13,18] and applications [13] to asymptotic complexity. Although not discussed in the paper, for applications, it is interesting to note that partial quasi-metric spaces can be considered as partially ordered topological spaces. This will be discussed in details elsewhere. In the paper, we will present the bicompletion of partial quasi-metric spaces in two parts: The first part will appeal to the use of partial metric spaces and the second part will appeal to the use of T0 -quasi metric spaces.
2 Partial quasi-metric spaces and T0 -quasi-metric spaces Definition 2.1 [8,10] Let X be a nonempty set. A map p : X × X → [0, ∞)
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