On bornology of extended quasi-metric spaces

Abstract

Beer studied the structure of sets equipped with the extended metrics with a focus on bornologies. In the paper [A. Piekosz and E. Wajch, Quazi-metrizability of bornological biuniverses inZF, J. Convex Anal. 2015], Piekosz and Wajch extended the well-known Hu's Theorem on boundedness in a topological space (see [S.-T. Hu, Boundedness in a topological space, J. Math. Pures Appl. 1949 ]) to the framework of quasi-metric spaces.In this note, we continue the work of Piekosz and Wajch. We show that many results on bornology of extended metric spaces due to Beer do not use the symmetry axiom of the extended metric, with appropriate modifications they still hold in the context of extended T 0 -quasi-metric spaces.Mathematics Subject Classification (2010). 54E35, 54E55, 54C35

Some Classes of Bounded Sets in Quasi-Metric Spaces

“…Recall from [3] that the bornology of quasi-pseudometric bounded sets is denoted by ( ) q X B . However, in [8], the family of totally bounded subsets and boubark bounded sets their bornologies are denoted by ( ) q X TB and ( ) q X BB respectively.…”

“…1 ⇒ 2: Since ( ) q X TB has a countable base by Hu's theorem (see ( [3], Theorem 4.18)) there exists an equivalent quasi-metric q′ such that ( ) ( ) 1 , whenever with 1 .…”

“…Morever, our recent work [3] has extended the concept of bornology from metric settings to the framework of quasi-metrics. Naturally, this has led to the speculation of what is the relationship between the bornology of bounded sets and other types of bornologies on quasi-metric spaces.…”

On Bourbaki-bounded sets on quasi-pseudometric spaces

“…Moreover, if q is an extended quasi-pseudometric on X (i.e., the distance between two points can be ∞), then a subset B of X can be included in D q (x, ϵ) for some x ∈ X, ϵ > 0 but its diameter diam(B) = sup{q(y, z) : y, z ∈ B} = ∞ (see [14, p. 2022 Let (X, q) be a quasi-pseudometric space. It has been observed in [9,11] that the collection B q (X) of all q-bounded subsets of X forms a bornology on X and this bornology is called the quasi-metric bornology determined by q. Furthermore, we have: B q s (X) = B q (X) and bornologies B q (X) and B q t (X) are equivalent.…”

Maps that preserve left (right) $K$-Cauchy sequences

“…It follows that * q (X) forms a bornology on X and this bornology is called the quasimetric bornology determined by q. Furthermore, We have the following observations instead of the one observed in [11] * q s (X) = * q (X)…”