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In this paper, we first revise some results and proofs on θ-metric spaces. Next, we construct an explicit metric to metrize a given θ-metric that gives an affirmative answer to an open question on the metrization of θ-metric spaces. After that, we use the obtained result to calculate such metric of known θ-metrics, and reprove a fixed point theorem in θ-metric spaces. [ABSTRACT FROM AUTHOR]

Extended suprametric spaces are defined, and the contraction principle is established using elementary properties of the greatest lower bound instead of the usual iteration procedure. Thereafter, some topological results and the Stone-type theorem are derived in terms of suprametric spaces. Also, we have shown that every suprametric space is metrizable. Further, we prove the existence of a solution of Ito-Doob type stochastic integral equations using our main fixed point theorem in extended suprametric spaces.

Extended suprametric spaces are defined, and the contraction principle is established using elementary properties of the greatest lower bound instead of the usual iteration procedure. Thereafter, some topological results and the Stone-type theorem are derived in terms of suprametric spaces. Also, we have shown that every suprametric space is metrizable. Further, we prove the existence of a solution of Ito-Doob type stochastic integral equations using our main fixed point theorem in extended suprametric spaces. [ABSTRACT FROM AUTHOR]

In two ways we introduce metrics on the set of all pseudoultrametrics, not exceeding a given compact pseudoultrametric on a fixed set, and prove that the obtained metrics are compact and topologically equivalent. To achieve this, we give a characterization of the sets being the hypographs of the mentioned pseudoultrametrics, and apply Hausdorff metric to their family. It is proved that the uniform convergence metric is a limit case of metrics defined via hypographs. It is shown that the set of all pseudoultrametrics, not exceeding a given compact pseudoultrametric, with the induced topology is a Lawson compact Hausdorff upper semilattice. [ABSTRACT FROM AUTHOR]

Given MM and NN Hausdorff topological spaces, we study topologies on the space C0(M;N){C}^{0}\left(M;\hspace{0.33em}N) of continuous maps f:M→Nf:M\to N. We review two classical topologies, the “strong” and the “weak” topology. We propose a definition of “mild topology” that is coarser than the “strong” and finer than the “weak” topology. We compare properties of these three topologies, in particular with respect to proper continuous maps f:M→Nf:M\to N, and affine actions when N=RnN={{\mathbb{R}}}^{n}. To define the “mild topology” we use “separation functions;” these “separation functions” are somewhat similar to the usual “distance function d(x,y)d\left(x,y)” in metric spaces (M,d)\left(M,d), but have weaker requirements. Separation functions are used to define pseudo balls that are a global base for a T2 topology. Under some additional hypotheses, we can define “set separation functions” that prove that the topology is T6. Moreover, under further hypotheses, we will prove that the topology is metrizable. We provide some examples of uses of separation functions: one is the aforementioned case of the mild topology on C0(M;N){C}^{0}\left(M;\hspace{0.33em}N). Other examples are the Sorgenfrey line and the topology of topological manifolds.

Given a smooth family of unparameterized curves such that through every point in every direction there passes exactly one curve, does there exist a Lagrangian with extremals being precisely this family? It is known that in dimension 2 the answer is positive. In dimension 3, it follows from the work of Douglas that the answer is, in general, negative. We generalise this result to all higher dimensions and show that the answer is actually negative for almost every such a family of curves, also known as path structure or path geometry. On the other hand, we consider path geometries possessing infinitesimal symmetries and show that path and projective structures with submaximal symmetry dimensions are variational. Note that the projective structure with the submaximal symmetry algebra, the so-called Egorov structure, is not pseudo-Riemannian metrizable; we show that it is metrizable in the class of Kropina pseudo-metrics and explicitly construct the corresponding Kropina Lagrangian. [ABSTRACT FROM AUTHOR]

Cheeger–Gromov finiteness results, asserting that there are only finitely many diffeomorphism types of manifolds satisfying certain geometric bounds, feature among the most prominent results in Riemannian geometry. To transplant those into Lorentzian geometry, one could use a functor between a Lorentzian and a Riemannian category, which, however, can be shown not to exist if the former contains Minkowski space and its isometries. Here, we construct a functor from a restricted category of Lorentzian manifolds-with-boundary (regions between two Cauchy surfaces) to a category of Riemannian manifolds-with-boundary that preserves geometric bounds and obtain, as a corollary, the first known Lorentzian Cheeger–Gromov type finiteness result. [ABSTRACT FROM AUTHOR]

The aim of this paper is to apply the diffusive metric technique defined by the spectral analysis of graph Laplacians to the set of the 41 cities belonging to AMBA, the largest urban concentration in Argentina, based on public transport and neighborhood. It could be expected that the propagation of any epidemic desease would follow the paths determined by those metrics. Our result reflects that the isolation measures decided by the health administration helped at the atenuation of the actual spread of COVID-19 in AMBA.

In this paper we introduce a metric on the space I(X) of idempotent probability measures on a given compact metric space (X; ρ), which extends the metric ρ. It is proven the introduced metric generates the pointwise convergence topology on I(X).

We identify two categories of quantale-valued convergence tower spaces that are isomorphic to the categories of quantale-valued metric spaces and quantale-valued partial metric spaces, respectively. As an application we state a quantale-valued metrization theorem for quantale-valued convergence tower groups.

This paper considers a model of a multidimensional space, in which variables are ordered and contain a finite number of values. Such a space may be analyzed with the n-tuple algebra developed by the authors as a natural combination of methods for the analysis of n-ary relations and logical structures. Some approaches enabling the application of some cluster analysis methods are proposed for this model. [ABSTRACT FROM AUTHOR]

In this paper we introduce a metric on the space I(X) of idempotent probability measures on a given compact metric space (X; ρ), which extends the metric ρ. It is proven the introduced metric generates the pointwise convergence topology on I(X). [ABSTRACT FROM AUTHOR]

This article is in line with earlier investigations done in [10–12,14,15,18] and several such works. Here our aim is to introduce and study two new completeness-like properties, namely, Bourbaki quasi-completeness and cofinally Bourbaki quasi-completeness (we use infinite chains instead of finite ones), which strictly lie between compactness and completeness, primarily in the setting of uniform spaces. We use the concept of finite-component covers [19] to define a new type of modification of a uniform space, which plays a crucial role throughout the paper. In Section 2, we relate cBq-completeness to the existing notion of superparacompactness [19]. Another significant and very natural problem we deal with is the topological problem of metrizability of a uniform space using a Bq-complete and a cBq-complete metric. We obtain results similar to the classical Čech theorem about the complete metrizability of a metric space X in terms of its Stone-Čech compactification βX , which are presented in sections 3 and 4 of the article. [ABSTRACT FROM AUTHOR]

In this paper we prove a metrization theorem on rectangular b -metric spaces. Then we get a sufficient and necessary condition for a rectangular b -metric space to be metrizable. [ABSTRACT FROM AUTHOR]

Quasimetric spaces have been an object of thorough investigation since Frink's paper appeared in 1937 and various generalisations of the axioms of metric spaces are now experiencing their well-deserved renaissance. The aim of this paper is to present two improvements of Frink's metrization theorem along with some fixed point results for single-valued mappings on quasimetric spaces. Moreover, Cantor's intersection theorem for sequences of sets which are not necessarily closed is established in a quasimetric setting. This enables us to give a new proof of a quasimetric version of the Banach Contraction Principle obtained by Bakhtin. We also provide error estimates for a sequence of iterates of a mapping, which seem to be new even in a metric setting. [ABSTRACT FROM AUTHOR]

In this paper, we give a simple counterexample to show again the limits of Frink's construction [17, page 134] and then use Frink's metrization technique to answer two conjectures posed by Berinde and Choban [5], and to calculate corresponding metrics induced by some b-metrics known in the literature. We also use that technique to prove a metrization theorem for 2-generalized metric spaces, and to deduce the Banach contraction principle in b-metric spaces and 2-generalized metric spaces from that in metric spaces. [ABSTRACT FROM AUTHOR]

We identify two categories of quantale-valued convergence tower spaces that are isomorphic to the categories of quantale-valued metric spaces and quantale-valued partial metric spaces, respectively. As an application we state a quantale-valued metrization theorem for quantale-valued convergence tower groups. [ABSTRACT FROM AUTHOR]

We investigate a space of weakly additive, order-preserving, normed and positively homogeneous functionals on a metric compact. We construct an analog of modified Kantorovich-Rubinshtein metric on a space O(X) of weakly additive normed functionals on a metric compact X. [ABSTRACT FROM AUTHOR]

It is a well-known fact that the topology induced by a fuzzy metric is metrizable. Nevertheless, the problem of how to obtain a classical metric from a fuzzy one in such a way that both induce the same topology is not solved completely. A new method to construct a classical metric from a fuzzy metric, whenever it is defined by means of an Archimedean t -norm, has recently been introduced in the literature. Motivated by this fact, we focus our efforts on such a method in this paper. We prove that the topology induced by a given fuzzy metric M and the topology induced by the metric constructed from M by means of such a method coincide. Besides, we prove that the completeness of the fuzzy metric space is equivalent to the completeness of the associated classical metric obtained by the aforementioned method. Moreover, such results are applied to obtain fuzzy versions of two well-known classical fixed point theorems in metric spaces, one due to Matkowski and the other one proved by Meir and Keeler. Although such theorems have already been adapted to the fuzzy context in the literature, we show an inconvenience on their applicability which motivates the introduction of these two new fuzzy versions. [ABSTRACT FROM AUTHOR]

In Geographic Information Systems (GIS), the exploration of the metric descriptions for topological spatial relations has been an active area of research. Construction processing of a metric description is directly influenced by spatial data model. Vector and raster data models are the two types of basic spatial data models. These two data models have complimentary advantages in terms of describing spatial relations between objects. The integrative data model of vector and raster stems from the integration of the advantages of vector and raster data model. Firstly, this paper defines qualitative topology relations by using the 9-intersection model. Secondly, the ratio of the grid number of intersection to the two objects, is used to determine the intersect component. Thirdly, the maximum and minimum distances are used to determine the closeness component. Finally, a triple group including qualitative topology relations, intersect component and closeness component, is proposed to describe topology spatial relation. Because of two advantages of integrative data model of vector and raster, the metric description of topology between different type objects can be realized more effectively in this paper. [ABSTRACT FROM AUTHOR]

In this paper, we first show that every b -metric space with the topology induced by its convergence is a semi-metrizable space and thus many properties of b -metric spaces used in the literature are obvious. Then, we prove the Stone-type theorem on b -metric spaces and get a sufficient condition for a b -metric space to be metrizable. We also give answers to the question posed by Khamsi and Hussain in [24] and discuss some results relating to the metrization of 2-metric spaces and S -metric spaces. [ABSTRACT FROM AUTHOR]

We discuss geometrical scenarios guaranteeing that functions defined on a given set may be extended to the entire ambient, with preservation of the class of regularity. This extends to arbitrary quasi-metric spaces work done by E.J. McShane in the context of metric spaces, and to geometrically doubling quasi-metric spaces work done by H. Whitney in the Euclidean setting. These generalizations are quantitatively sharp. [ABSTRACT FROM AUTHOR]

Let X, n ∈ ℤ be a sequence of non-empty sets, ψ: X → ℝ. We consider the relation E = E(( X, ψ)) on Π X by ( x, y) ∈ E(( X, ψ)) ⇔ Σ ψ( x( n), y( n)) < + ∞. If E is an equivalence relation and all ψ, n ∈ ℤ, are Borel, we show a trichotomy that either ℝ/ ℓ ⩽ E, E ⩾ E, or E ⩾ E. We also prove that, for a rather general case, E(( X, ψ)) is an equivalence relation iff it is an ℓ-like equivalence relation. [ABSTRACT FROM AUTHOR]

Abstract: We study spaces with sharp bases and bases of countable order. A characterization of spaces with external bases of countable order is established (Theorem 2.7). Some necessary and sufficient conditions for a space , where S is the convergent sequence, to have a sharp base are given (Theorem 3.2). It follows that a pseudocompact space X is metrizable iff has a sharp base (Corollary 3.3). It is proved that a sharp base of finite rank is a uniform base (Theorem 4.4). Some other new results are also obtained, and some open questions are formulated. [Copyright &y& Elsevier]

Abstract: This is a survey article on the work of Professor Jun-iti Nagata. We present his educational and professional career, his activities and contributions to the topology community and his work on topology, especially the theory of uniform spaces, rings of continuous functions, metrization and generalized metric spaces. [Copyright &y& Elsevier]

Abstract: Given a space M, a family of sets of a space X is ordered by M if { is a compact subset of M} and implies . We study the class of spaces which have compact covers ordered by a second countable space. We prove that a space belongs to if and only if it is a Lindelöf Σ-space. Under , if X is compact and has a compact cover ordered by a Polish space then X is metrizable; here is the diagonal of the space X. Besides, if X is a compact space of countable tightness and belongs to then X is metrizable in ZFC. We also consider the class of spaces X which have a compact cover ordered by a second countable space with the additional property that, for every compact set there exists with . It is a ZFC result that if X is a compact space and belongs to then X is metrizable. We also establish that, under CH, if X is compact and belongs to then X is countable. [ABSTRACT FROM AUTHOR]

Abstract: This paper deals with the study of closed images or quasi-perfect images of Nagata spaces, contraconvergent spaces, weak contraconvergent spaces, ks-spaces, γ-spaces and wγ-spaces and, of metrization theorems involving these spaces. We prove that the closed images of contraconvergent (weak contraconvergent) spaces are contraconvergent (weak contraconvergent) and that quasi-perfect images of γ- (wγ-)spaces are γ (wγ). [Copyright &y& Elsevier]

Abstract: In this paper we examine the role of the β-space property (equivalently of the MCM-property) in generalized ordered (GO-)spaces and, more generally, in monotonically normal spaces. We show that a GO-space is metrizable iff it is a β-space with a -diagonal and iff it is a quasi-developable β-space. That last assertion is a corollary of a general theorem that any β-space with a σ-point-finite base must be developable. We use a theorem of Balogh and Rudin to show that any monotonically normal space that is hereditarily monotonically countably metacompact (equivalently, hereditarily a β-space) must be hereditarily paracompact, and that any generalized ordered space that is perfect and hereditarily a β-space must be metrizable. We include an appendix on non-Archimedean spaces in which we prove various results announced without proof by Nyikos. [Copyright &y& Elsevier]

We give a new metrization theorem on terms of a new structure introduced by the authors in [Rend. Instit. Mat. Univ. Trieste 30 (1999) 21–30] and called fractal structure. This allows us to approach some classical and new metrization theorems (due to Nagata, Smirnov, Moore, Arhangel''skii, Frink, Borges, Hung, Morita, Fletcher, Lindgren, Williams, Collins, Roscoe, Reed, Rudin, Hanai, Stone, Burke, Engelking and Lutzer) from a new point of view. [Copyright &y& Elsevier]

In this paper we study the role of Stepanova''s continuous separating families in the class of linearly ordered and generalized ordered spaces and we construct examples of paracompact spaces that have strong base properties (such as point-countable bases or σ-disjoint bases), have continuous separating families, and yet are non-metrizable. [Copyright &y& Elsevier]

We continue Gartside, Moody, and Stares' study of versions of monotone paracompactness. We show that the class of spaces with a monotone open closure-preserving operator is strictly larger than those with a monotone open locally-finite operator. We prove that monotonically metacompact GO-spaces have a monotone open star-finite operator, and so do GO-spaces with a monotone (open or not) closure-preserving operator, whose underlying LOTS has a σ -closed-discrete dense subset. A GO-space with a σ -closed-discrete dense subset and a monotone closure-preserving operator is metrizable. A compact LOTS with a monotone open closure-preserving operator is metrizable. [ABSTRACT FROM AUTHOR]

We study the Voronoi diagrams of a finite set of Cauchy distributions and their dual complexes from the viewpoint of information geometry by considering the Fisher-Rao distance, the Kullback-Leibler divergence, the chi square divergence, and a flat divergence derived from Tsallis entropy related to the conformal flattening of the Fisher-Rao geometry. We prove that the Voronoi diagrams of the Fisher-Rao distance, the chi square divergence, and the Kullback-Leibler divergences all coincide with a hyperbolic Voronoi diagram on the corresponding Cauchy location-scale parameters, and that the dual Cauchy hyperbolic Delaunay complexes are Fisher orthogonal to the Cauchy hyperbolic Voronoi diagrams. The dual Voronoi diagrams with respect to the dual flat divergences amount to dual Bregman Voronoi diagrams, and their dual complexes are regular triangulations. The primal Bregman Voronoi diagram is the Euclidean Voronoi diagram and the dual Bregman Voronoi diagram coincides with the Cauchy hyperbolic Voronoi diagram. In addition, we prove that the square root of the Kullback-Leibler divergence between Cauchy distributions yields a metric distance which is Hilbertian for the Cauchy scale families. [ABSTRACT FROM AUTHOR]

In this paper, we study two questions of A. Petruşel and G. Petruşel in b-metric fixed point theory [12]. The main results of the paper are as follows.Using the b-metric metrization theorem [9], fixed point results in the setting of b-metric spaces proved in [10-12] and some others may be seen as consequences of Ran-Reurings fixed point theorem in the classical metric spaces [13, Theorem 2.1]. This gives a partial answer to the question in [12, Remark 3.(2)].Using the product space of two JS-metric spaces, main results of [12] and some others in the setting of b-metric spaces can be extended to the setting of JS-spaces. This answers the question in [12, Open question on page 1809]. [ABSTRACT FROM AUTHOR]