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Motivated by the recent introduction of the intrinsic semilattice entropy, we study generalized quasi-metric semilattices and their categories. We investigate the relationship between these objects and generalized semivaluations, extending Nakamura and Schellekens' approach. Finally, we use this correspondence to compare the intrinsic semilattice entropy and the semigroup entropy induced in particular situations, like sets, torsion abelian groups and vector spaces.

In this paper, we discuss some topological properties of graphical metric spaces and introduce the G-set metric with respect to a graphical metric. Some fixed point results are introduced which generalize the famous Nadler’s fixed point theorem.

In this study, at first we prove that the existence of best proximity points for cyclic nonexpansive mappings is equivalent to the existence of best proximity pairs for noncyclic nonexpansive mappings in the setting of strictly convex Banach spaces by using the projection operator. In this way, we conclude that the main result of the paper [Proximal normal structure and nonexpansive mappings, Studia Math. 171 (2005), 283–293] immediately follows. We then discuss the convergence of best proximity pairs for noncyclic contractions by applying the convergence of iterative sequences for cyclic contractions and show that the convergence method of a recent paper [Convergence of Picard's iteration using projection algorithm for noncyclic contractions, Indag. Math. 30 (2019), no. 1, 227–239] is obtained exactly from Picard’s iteration sequence.

Related to earlier work on existence results of best proximity points (pairs) for cyclic (noncyclic) condensing operators of integral type in the setting of reflexive Busemann convex spaces, in this paper, we introduce another class of cyclic (noncyclic) condensing operators and study the existence of best proximity points (pairs) as well as coupled best proximity points (pairs) in such spaces. Then, we present an application of our main existence result to study the existence of an optimal solution for a system of differential equations.

Let (A, B) be a nonempty, closed and convex pair in a reflexive and Busemann convex space X, and $$(E,F)\subseteq (A,B)$$ be a nonempty and proximinal pair in X such that $$\mathrm{dist}(E,F)=\mathrm{dist}(A,B)$$. We prove that the pair $$(\overline{\mathrm{con}}(E),\overline{\mathrm{con}}(F))$$ is also proximinal, where $$\overline{\mathrm{con}}(E)$$ denotes the closed convex hull of the set E. Moreover, we introduce a new notion of cyclic (noncyclic) mappings involving measure of noncompactness and obtain some new existence results of best proximity points (pairs). As an application of our main conclusions, we study the existence of an optimal solution for a system of integrodifferential equations under new sufficient conditions.

H.J.K. Junnila [9] called a neighbournet N on a topological space X unsymmetric provided that for each x , y ∈ X with y ∈ ( N ∩ N − 1 ) ( x ) we have that N ( x ) = N ( y ) . Motivated by this definition, we shall call a T 0 -quasi-metric d on a set X unsymmetric provided that for each x , y , z ∈ X the following variant of the triangle inequality holds: d ( x , z ) ≤ d ( x , y ) ∨ d ( y , x ) ∨ d ( y , z ) . Each T 0 -ultra-quasi-metric is unsymmetric. We also note that for each unsymmetric T 0 -quasi-metric d, its symmetrization d s = d ∨ d − 1 is an ultra-metric. Furthermore we observe that unsymmetry of T 0 -quasi-metrics is preserved by subspaces and suprema of nonempty finite families, but not necessarily under conjugation. In addition we show that the bicompletion of an unsymmetric T 0 -quasi-metric is unsymmetric. The induced T 0 -quasi-metric of an asymmetrically normed real vector space X is unsymmetric if and only if X = { 0 } . Our results are illustrated by various examples. We also explain how our investigations relate to the theory of ordered topological spaces and questions about (pairwise) strong zero-dimensionality in bitopological spaces.

We continue the study of asymmetry in $T_0$-quasi-metric spaces. In this context we introduce the property of symmetric connectedness for a $T_0$-quasi-metric space. We present some methods in order to find the symmetrically connected pairs of a $T_0$-quasi-metric space. We also show that the problem to determine the symmetry components of points turns out to be easier when formulated for the induced $T_0$-quasi-metric of an asymmetrically normed real vector space. In addition, as a kind of opposite to the notion of a metric space, we define antisymmetric $T_0$-quasi-metric spaces. Subsequently some useful results about antisymmetry can be emphasized by describing the property of antisymmetric connectedness for a $T_0$-quasi-metric space. Finally, we observe that there are natural relations between the theory of (anti)symmetrically connected $T_0$-quasi-metric spaces and the theory of connectedness in the sense of graph theory.

In the setting of convex metric spaces, we introduce the two geometric notions of uniform convexity in every direction as well as sequential convexity. They are used to study a concept of proximal normal structure. We also consider the class of noncyclic relatively nonexpansive mappings and analyze the min-max property for such mappings. As an application of our main results we conclude with some best proximity pair theorems for noncyclic mappings.

In this paper we continue the investigations of the relationship between T 0 -quasi-metric spaces and partially ordered metric spaces. Among other things, we establish an equivalence of categories between so-called maximal T 0 -quasi-metric spaces and partially ordered metric spaces produced by T 0 -quasi-metrics. In the linear context we give geometric interpretations of the obtained results. In particular, we also derive a representation theorem for injective asymmetrically normed spaces.

Given a T 0 -ultra-quasi-metric u on a set X, we write u s for its symmetrization u ∨ u − 1 . In this paper we show that there exists a T 0 -ultra-quasi-metric v on X such that v ≤ u , v s = u s and the specialization order of v is linear. We also discuss connections of this statement with related results about Robinsonian dissimilarities and orderability of ultra-metrizable topological spaces.

An explicit description of the algebraic and vector lattice operations on the Isbell-convex hull of an asymmetrically normed real vector space is provided. Connections between the concepts of Isbell-convexity, injectivity and Dedekind-completeness of asymmetrically normed real vector spaces are also considered.

Let $${(X, m, \leq)}$$ be a partially ordered metric space, that is, a metric space (X, m) equipped with a partial order $${\leq}$$ on X. We say that a T 0-quasi-metric d on X is m-splitting provided that $${d\vee d^{-1}=m}$$ . Furthermore d is said to be $${(X, m, \leq)}$$ -producing provided that d is m-splitting and the specialization partial preorder of d is equal to $${\leq}$$ . It is known and easy to see that if $${(X, m, \leq)}$$ is a partially ordered metric space that is produced by a T 0-quasi-metric and $${\leq}$$ is a total order, then there exists exactly one producing T 0-quasi-metric on X. We first will give an example that shows that a partially ordered metric space can be uniquely produced by a T 0-quasi-metric although $${\leq}$$ is not total. Then we present solutions to the following two problems: Let $${(X, m, \leq)}$$ be a partially ordered metric space and A a subset of X. (1) Suppose that d is a T 0-quasi-metric on A which is $${m\vert ({A \times A})}$$ -splitting. When can d be extended to an m-splitting T 0-quasi-metric $${\widetilde{d}}$$ on X? (2) Suppose that d is a T 0-quasi-metric on A which is $${(A, m\vert (A\times A)}$$ , $${{\leq}\vert ({A\times A}))}$$ -producing. When can d be extended to a T 0-uasi-metric $${\widetilde{d}}$$ on X that produces $${(X, m, \leq)?}$$

We continue our investigation on those (partially) ordered metric spaces ( X , m , ≤ ) for which there exists a T 0 -quasi-metric d on X such that sup ⁡ { d , d − 1 } = m and for any x , y ∈ X we have that x ≤ y if and only if d ( x , y ) = 0 . In particular such partially ordered metric spaces are determined (in the sense of Nachbin) by a T 0 -quasi-uniformity with a countable base. We show that the compatibility conditions obtained in an earlier study between partial order and metric on the investigated spaces X become more transparent when additional compatibility conditions between metric and partial order with appropriate algebraic operations on X are assumed to hold.

The authors define and investigate convexity structures in the sense of Takahashi in T 0 -quasi-metric spaces. They prove that numerous important results about convexity structures in metric spaces can be generalized to the quasi-metric setting. They also show that the latter convexity structures naturally occur in asymmetrically normed real vector spaces and in q-hyperconvex T 0 -quasi-metric spaces. In the T 0 -quasi-metric setting they explore various interesting additional conditions that convexity structures in the sense of Takahashi can satisfy.

It is well known that by Szpilrajn's Theorem each partial order on a set can be extended to a linear order. In this paper we study variants of that theorem that hold for T 0 -quasi-metrics instead of partial orders. More specifically, given a metric space ( X , m ) , we call a T 0 -quasi-metric d defined on the set X m-splitting provided that its symmetrization d s = d ∨ d − 1 is equal to m. For a given metric space ( X , m ) we obtain results about T 0 -quasi-metrics on X that are minimal among the collection of all m-splitting T 0 -quasi-metrics on X. We are also interested in the maximal possible specialization orders of m-splitting T 0 -quasi-metrics.

It is known that if K is a compact subset of the (separable complete) metric Urysohn space (${\mathbb U}$, d) and f is a Katětov function on the subspace K of (${\mathbb U}$, d), then there is z ∈ ${\mathbb U}$ such that d(z, x) = f(x) for all x ∈ K.Answering a question of Normann, we show in this article that the supseparable bicomplete q-universal ultrahomogeneous T0-quasi-metric space (q${\mathbb U}$, D) recently discussed by the authors satisfies a similar property for Katětov function pairs on subsets that are compact in the associated metric space (q${\mathbb U}$, Ds).

We construct a T1-space that is not hereditarily compact, although each of its open sets is the intersection of two compact open sets. The search for such a space was motivated by a problem in the theory of quasi-proximities.

A topological space is called a uqu space [10] if it admits a unique quasi-uniformity. Answering a question [2, Problem B, p. 45] of P. Fletcher and W. F. Lindgren in the affirmative we show in [8] that a topological space X is a uqu space if and only if every interior-preserving open collection of X is finite. (Recall that a collection ℒ of open sets of a topological space is called interior-preserving if the intersection of an arbitrary subcollection of ℒ is open (see e.g. [2, p. 29]).) The main step in the proof of this result in [8] shows that a topological space in which each interior-preserving open collection is finite is a transitive space. (A topological space is called transitive (see e.g. [2, p. 130]) if its fine quasi-uniformity has a base consisting of transitive entourages.) In the first section of this note we prove that each hereditarily compact space is transitive. The result of [8] mentioned above is an immediate consequence of this fact, because, obviously, a topological space in which each interior-preserving open collection is finite is hereditarily compact; see e.g. [2, Theorem 2.36]. Our method of proof also shows that a space is transitive if its fine quasi-uniformity is quasi-pseudo-metrizable. We use this result to prove that the fine quasi-uniformity of a T1 space X is quasi-metrizable if and only if X is a quasi-metrizable space containing only finitely many nonisolated points. This result should be compared with Proposition 2.34 of [2], which says that the fine quasi-uniformity of a regular T1 space has a countable base if and only if it is a metrizable space with only finitely many nonisolated points (see e.g. [11] for related results on uniformities). Another by-product of our investigations is the result that each topological space with a countable network is transitive.

We study some aspects of the space $QPM(X)$of all quasi-pseudometrics on a set $X$equipped with the extended $T_0$-quasi-metric $A_X(f,g)=\sup_{(x,y)\in X\times X}(f(x,y)-g(x,y))$ whenever $f,g\in QPM(X)$.We observe that this space is bicomplete and exhibit various closedsubspaces of $( QPM(X), \tau((A_X)^s))$.In the second part of the paper, as a rough way to measure the asymmety of a quasi-pseudometric $f$ on a set $X$, we investigate some properties of the value $(A_X)^s(f,f^{-1}).

We present a construction of the B -completion of a T 0 -quasi-metric space ( X , d ) that is based on the injective hull of ( X , d ) . The B -completion of a T 0 -quasi-metric space ( X , d ) is a T 0 -quasi-metric extension of ( X , d ) studied by H.-P.A. Kunzi and C. Makitu Kivuvu. In general, the B -completion of a T 0 -quasi-metric space is larger than the better known bicompletion of ( X , d ) . In the special case of a balanced T 0 -quasi-metric space the B -completion coincides with a completion that was first investigated by D. Doitchinov. Among other things, in this note we show that the B -completion of the natural T 0 -quasi-metric of a partially ordered set (possessing a smallest and a largest element) can be identified with the Dedekind–MacNeille completion of that partially ordered set.

In his well-known paper dealing with the construction of the injective hull of a metric space Isbell introduced the concept of an endpoint of a (compact) metric space. In the present article we introduce similarly the notion of an endpoint in a (joincompact) T 0 -quasi-metric space. It turns out that in a (joincompact) T 0 -quasi-metric space there is a dual concept which we shall call a startpoint. With the help of these concepts we are able to generalize some of the classical results on endpoints in metric spaces to the quasi-metric setting.

In this paper we present some results on selection properties in asymmetric generalized metric and uniform spaces. We demonstrate differences between selection properties of these spaces and selection properties of metric and uniform spaces.

We continue our work on endpoints and startpoints inT0-quasimetric spaces. In particular we specialize some of our earlier results to the case of two-valuedT0-quasimetrics, that is, essentially, to partial orders. For instance, we observe that in a complete lattice the startpoints (resp., endpoints) in our sense are exactly the completely join-irreducible (resp., completely meet-irreducible) elements. We also discuss for a partially ordered set the connection between its Dedekind-MacNeille completion and theq-hyperconvex hull of its naturalT0-quasimetric space.

We study a concept of hyperconvexity that is appropriate to the category of T 0 -quasi-metric spaces (called di-spaces in the following) and nonexpansive maps. An explicit construction of the corresponding hull (called Isbell-convex hull or, more briefly, Isbell-hull) of a di-space is provided.

[EN] The authors study quasi-uniformities that are generated by a family of weightable quasi-pseudometrics. Each totally bounded quasi-uniformity is of this kind. In some sense, which is described in this article, a weightable quasi-uniformity is fairly symmetric, with the associated weights generating small symmetrizers. In the second part of the article we continue our investigations by generalizing a subclass of weightable quasi-uniformities to a more abstract level. We introduce the concept of a t-symmetrizable quasi-uniformity, that is, a quasi-uniformity possessing the property that there exists a totally bounded quasi-uniformity such that is a uniformity. It turns out that t-symmetrizable quasi-uniformities are closely related to quasi-uniformities generated by weightable quasi-pseudometrics possessing bounded weight functions. We show that several results that were originally proved for weightable quasi-pseudometrics (with bounded weights) still hold in a such apparently broader context., The first author would like to thank the South African National Research Foundation for partial financial support. The second author thanks the support of the Spanish Ministry of Science and Innovation, grant MTM2009-12872-C02-01.

Jointly with E. Kemajou, in previous work we constructed the so-called Isbell-hull of a T0-quasi-metric space. In this article we continue these investigations by presenting a similar construction in the category of T0-ultra-quasi-metric spaces and contracting maps. Comparable studies in the area of ultra-metric spaces have been conducted before by Bayod and Martinez-Maurica.