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[EN] In the context of fuzzy metrics in the sense of George and Veeramani, we introduce the concept of stratified fuzzy metric. Many well-known fuzzy metrics are stratified. We prove that stratified strong fuzzy metric spaces (X, M, *) are completable, under the assumption that * is integral (positive). In particular, stratified fuzzy ultrametric spaces are completable. (C) 2017 Elsevier B.V. All rights reserved., Valentin Gregori acknowledges the support of the Spanish Ministry of Economy and Competitiveness under Grant MTM2015-64373-P.

For a metric space ( X , d ) , a subset A resolves ( X , d ) if each point x ∈ X is uniquely determined by the distances d ( x , a ) for a ∈ A . Also the metric dimension of ( X , d ) is the smallest cardinality m d ( X ) such that there is a set A of the cardinality m d ( X ) that resolves X. In this note we are going to determine the metric dimension of metric orbit spaces in some special cases and find an upper bound for a general case. This category contains a vast domain of topological spaces and topological manifolds.

For a family of groups G λ which are equipped with bi-invariant metrics, we construct, in a very explicit manner, a bi-invariant metric on the free product of the G λ . The universal properties of this construction are investigated and some open questions are formulated.

In his well-known paper dealing with metric trees and tight extension of metric spaces, Dress studied the relationship between metric trees and hyperconvex metric spaces. In the present work, we introduce similarly the notion of quasi-metric tree of a T 0 -quasi-metric space. In particular, we show that large parts of the theory of metric trees do not use the symmetry of the metric and, under appropriate modifications, it still holds essentially unchanged for T 0 -quasi-metrics.

A simplified variant of the definition of the completeness for metric mappings (that is closed to the standard definition of the completeness for metric spaces) is obtained. This result is used to construct the completions of metric mappings by a method close to the standard completion method for metric spaces. Relations between completions, fibrewise completions and fibrewise complete extensions of metric mappings are clarified. It is shown that for closed metric mappings all of these extensions coincide (since the completeness of these mappings coincides with their fibrewise completeness). The Lavrentieff theorem (about G δ -extensions of homeomorphisms between subsets of metric spaces) is extended to metric mappings. It is proved that a uniformly continuous map-morphism of metric mappings may be extended to a uniformly continuous map-morphism of their completions. The end of the paper contains generalizations of the Nagata–Smirnov and Bing metrization theorems for mappings.

In the paper for any n ∈ ω we construct a separable complete metric n-dimensional space containing isometrically all compact metric n-dimensional spaces.

Recently, Suzuki [T. Suzuki, A generalized Banach contraction principle that characterizes metric completeness, Proc. Amer. Math. Soc. 136 (2008) 1861–1869] proved a fixed point theorem that is a generalization of the Banach contraction principle and characterizes the metric completeness. In this paper we prove an analogous fixed point result for a self-mapping on a partial metric space or on a partially ordered metric space. Our results on partially ordered metric spaces generalize and extend some recent results of Ran and Reurings [A.C.M. Ran, M.C. Reurings, A fixed point theorem in partially ordered sets and some applications to matrix equations, Proc. Amer. Math. Soc. 132 (2004) 1435–1443], Nieto and Rodriguez-Lopez [J.J. Nieto, R. Rodriguez-Lopez, Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations, Order 22 (2005) 223–239]. We deduce, also, common fixed point results for two self-mappings. Moreover, using our results, we obtain a characterization of partial metric 0-completeness in terms of fixed point theory. This result extends Suzukiʼs characterization of metric completeness.

We have shown in this paper that a (complete) cone metric space ( X , E , P , d ) is indeed (completely) metrizable for a suitable metric D. Moreover, given any finite number of contractions f 1 , … , f n on the cone metric space ( X , E , P , d ) , D can be defined in such a way that these functions become also contractions on ( X , D ) .

We prove some new results on the classification of metrics. These types of metrics include the ultrametrics, the additive metrics, the L p -embeddable metrics, the hypermetrics, the metrics of negative type and the metrics with one positive eigenvalue. We pose many open questions.

If (Y, d) is a complete metric space with a non-Archimedean metric d, then there exists a selection on the set F (Y) of all nonempty closed subsets of Y which is continuous with respect to both the Vietoris and Hausdorff topologies on F (Y) .

We give some results about o-extensions in the sense of Klebanov for classes of generalized metric spaces.