scholarly journals Two theorems on generalised metric spaces

1997 ◽  
Vol 55 (2) ◽  
pp. 197-206
Author(s):  
Sergey Svetlichny
Keyword(s):  
Compact Space ◽  
Distance Function ◽  
Weak Topology ◽  
Metric Spaces ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Compact Spaces ◽  
Countably Compact Space ◽  
Countably Compact ◽  
Necessary And Sufficient

We prove that any compact space, and even any countably compact space having the weak topology with respect to a sequence of symmetrisable subspaces, is metrisable. This generalises results of Arhangel'skii and Nedev on metrisability of symmetrisable compact spaces. Also we define and study contraction functions on generalised metric spaces whose topology can be described in terms of a ‘distance function’ which is not quite a metric. In particular we present necessary and sufficient conditions for a space of countable pseudo-character to be submetrisable in terms of real-valued contraction functions on this space.

scholarly journals Maximal topologies

1973 ◽  
Vol 15 (3) ◽  
pp. 279-290 ◽  
Author(s):  
Asit Baran Raha
Keyword(s):  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
The Other ◽  
Compact Spaces ◽  
Other Hand ◽  
Countably Compact ◽  
Countably Compact Spaces ◽  
Necessary And Sufficient

This article is devoted to studying maximal π spaces where π = Lindelöf, countably compact, connected, lightly compact or pseudocompact. Necessary and sufficient conditions for Lindelöf or countably compact spaces to be maximal Lindelöf or maximal countably compact have been obtained. On the other hand only necessary conditions for maximal π spaces have been deduced where π = connected, lightly compact or pseudocompact.

scholarly journals Further results on images of metric spaces

2015 ◽  
Vol 98 (112) ◽  
pp. 179-191
Author(s):  
Van Dung
Keyword(s):  
Metric Space ◽  
Metric Spaces ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Weakly Continuous ◽  
Locally Separable Metric Spaces ◽  
Necessary And Sufficient ◽  
Separable Metric Space

We introduce the notion of an ls-?-Ponomarev-system to give necessary and sufficient conditions for f:(M,M0) ? X to be a strong wc-mapping (wc-mapping, wk-mapping) where M is a locally separable metric space. Then, we systematically get characterizations of weakly continuous strong wc-images (wc-images, wk-images) of locally separable metric spaces by means of certain networks. Also, we give counterexamples to sharpen some results on images of locally separable metric spaces in the literature.

On Weak Asymptotic Periodicity

2020 ◽  
Vol 30 (02) ◽  
pp. 2050030
Author(s):  
Karol Gryszka
Keyword(s):  
Dynamical Systems ◽  
Asymptotic Property ◽  
Metric Spaces ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Topological Conjugacy ◽  
Asymptotic Periodicity ◽  
Necessary And Sufficient

We introduce the asymptotic property associated with recurrence-like behavior of orbits in dynamical systems in general metric spaces. We define a notion of weak asymptotic periodicity and determine its elementary properties and relations including the invariance by topological conjugacy. We use the equicontinuity and the topology of the space to describe necessary and sufficient conditions for the existence of such a behavior.

Characterizations of Spherical Neighbourhoods

1970 ◽  
Vol 22 (2) ◽  
pp. 431-435 ◽  
Author(s):  
C. M. Petty ◽  
J. M. Crotty
Keyword(s):  
Metric Space ◽  
Metric Spaces ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Characterization Problem ◽  
Necessary And Sufficient

If Σ is a specified class of metric spaces and M ∈ Σ, then the characterization problem is to find necessary and sufficient conditions which distinguish the spherical neighbourhoods (open spheres) of M among a specified class of subsets of M.In a metric space M the notation pqr means p ≠ q ≠ r and pq + qr = pr.M is said to be uniformly locally externally convex if there exists δ > 0 such that if p, q ∈ M, p ≠ q, and pq < δ, then there exists r ∈ M such that the relation pqr subsists. We will prove the following result.

scholarly journals Common stationary points of multivalued mappings on bounded metric spaces

2000 ◽  
Vol 24 (11) ◽  
pp. 773-779 ◽  
Author(s):  
Zeqing Liu ◽  
Shin Min Kang
Keyword(s):  
Stationary Point ◽  
Metric Spaces ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Stationary Points ◽  
Multivalued Mappings ◽  
Necessary And Sufficient

Necessary and sufficient conditions for the existence of common stationary points of two multivalued mappings and common stationary point theorems for multivalued mappings on bounded metric spaces are given. Our results extend the theorems due to Fisher in 1979, 1980, and 1983 and Ohta and Nikaido in 1994.

scholarly journals On Conditions for Unrectifiability of a Metric Space

2014 ◽  
Vol 3 (1) ◽  
Author(s):  
Piotr Hajłasz ◽  
Soheil Malekzadeh
Keyword(s):  
Metric Space ◽  
Implicit Function Theorem ◽  
Metric Spaces ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Implicit Function ◽  
Heisenberg Groups ◽  
Lipschitz Map ◽  
Necessary And Sufficient

Abstract We find necessary and sufficient conditions for a Lipschitz map f : E ⊂ ℝk → X into a metric space to satisfy ℋk(f(E)) = 0. An interesting feature of our approach is that despite the fact that we are dealing with arbitrary metric spaces, we employ a variant of the classical implicit function theorem. Applications include pure unrectifiability of the Heisenberg groups.

scholarly journals Uniqueness of spaces pretangent to metric spaces at infinity

2019 ◽  
Vol 16 (1) ◽  
pp. 57-87
Author(s):  
Oleksiy Dovgoshey ◽  
Victoria Bilet
Keyword(s):  
Metric Space ◽  
Real Line ◽  
Metric Spaces ◽  
Sufficient Conditions ◽  
Logarithmic Spiral ◽  
Necessary And Sufficient Conditions ◽  
Scaling Sequence ◽  
Necessary And Sufficient

We find the necessary and sufficient conditions under which an unbounded metric space \(X\) has, at infinity, a unique pretangent space \(\Omega^{X}_{\infty,\tilde{r}}\) for every scaling sequence \(\tilde{r}\). In particular, it is proved that \(\Omega^{X}_{\infty,\tilde{r}}\) is unique and isometric to the closure of \(X\) for every logarithmic spiral \(X\) and every \(\tilde{r}\). It is also shown that the uniqueness of pretangent spaces to subsets of a real line is closely related to the ''asymptotic asymmetry'' of these subsets.

scholarly journals I-LOCALIZED SEQUENCES IN METRIC SPACES

2020 ◽  
pp. 459
Author(s):  
Anar Adiloğlu Nabiev ◽  
Ekrem Savaş ◽  
Mehmet Gürdal
Keyword(s):  
Metric Spaces ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Cauchy Sequences ◽  
Necessary And Sufficient

In this paper we have introduced the I-localized and the I^{∗}-localized sequences in metric spaces and investigate some basics properties of the I-localized sequences related with I-Cauchy sequences. Also we have obtained some necessary and sufficient conditions for the I-localized sequences to be an I-Cauchy sequences. It is also defined uniformly the I-localized sequences on metric spaces and its relation with I-Cauchy sequences has been  obtained.

scholarly journals On Topological Properties of Metrics Defined via Generalized “Linking Construction”

2017 ◽  
Vol 2017 ◽  
pp. 1-7
Author(s):  
Marcin Borkowski ◽  
Dariusz Bugajewski ◽  
Adam Burchardt
Keyword(s):  
Metric Spaces ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Topological Properties ◽  
Necessary And Sufficient

We analyze topological properties of metric spaces obtained by using Száz’s construction, which we used to call generalized “linking construction.” In particular, we provide necessary and sufficient conditions for completeness of metric spaces obtained in this way. Moreover, we examine the relation between Száz’s construction and the “linking construction.” A particular attention is drawn to the “floor” metric, the analysis of which provides some interesting observations.

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