Necessary and sufficient conditions in order that a metric space can be uniformly retracted into each of its nonempty closed sets

1971 ◽  
Vol 12 (1) ◽  
pp. 162-164
Author(s):  
G. L. Garg
Keyword(s):  
Metric Space ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Closed Sets ◽  
Necessary And Sufficient

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.

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 Fixed point theorems for non-self maps I

1994 ◽  
Vol 17 (4) ◽  
pp. 713-716 ◽  
Author(s):  
Troy L. Hicks ◽  
Linda Marie Saliga
Keyword(s):  
Fixed Point ◽  
Metric Space ◽  
Closed Subset ◽  
Fixed Point Theorems ◽  
Complete Metric Space ◽  
Sufficient Conditions ◽  
General Setting ◽  
Necessary And Sufficient Conditions ◽  
Necessary And Sufficient

Supposef:C→XwhereCis a closed subset ofX. Necessary and sufficient conditions are given forfto have a fixed point. All results hold whenXis complete metric space. Several results hold in a much more general setting.

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 On characterizations of fixed points

2001 ◽  
Vol 27 (7) ◽  
pp. 391-397 ◽  
Author(s):  
Zeqing Liu ◽  
Lili Zhang ◽  
Shin Min Kang
Keyword(s):  
Fixed Points ◽  
Metric Space ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Equivalent Condition ◽  
Compact Mapping ◽  
Necessary And Sufficient

We give some necessary and sufficient conditions for the existence of fixed points of a family of self mappings of a metric space and we establish an equivalent condition for the existence of fixed points of a continuous compact mapping of a metric space.

scholarly journals Free Cells in Hyperspaces of Graphs

Mathematics ◽  
2021 ◽  
Vol 9 (14) ◽  
pp. 1627
Author(s):  
José Ángel Juárez Morales ◽  
Gerardo Reyna Hernández ◽  
Jesús Romero Valencia ◽  
Omar Rosario Cayetano
Keyword(s):  
Metric Space ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Free Cells ◽  
Arcwise Connected ◽  
Necessary And Sufficient

Often for understanding a structure, other closely related structures with the former are associated. An example of this is the study of hyperspaces. In this paper, we give necessary and sufficient conditions for the existence of finitely-dimensional maximal free cells in the hyperspace C(G) of a dendrite G; then, we give necessary and sufficient conditions so that the aforementioned result can be applied when G is a dendroid. Furthermore, we prove that the arc is the unique arcwise connected, compact, and metric space X for which the anchored hyperspace Cp(X) is an arc for some p∈X.

Characterizations of invariant distributions

1985 ◽  
Vol 97 (2) ◽  
pp. 349-355
Author(s):  
Timothy C. Brown ◽  
Donald I. Cartwright ◽  
G. K. Eagleson
Keyword(s):  
Metric Space ◽  
Random Element ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Invariant Distributions ◽  
Random Transformation ◽  
Necessary And Sufficient ◽  
Separable Metric Space

AbstractLet (S, ρ) be a separable metric space and G a group of transformations of S. Necessary and sufficient conditions for a distribution on S to be invariant under G are derived in terms of the behaviour of the convolution of a random transformation from G and a random element of S.

scholarly journals Asymptotic Properties of -Expansive Homeomorphisms on a Metric -Space

2012 ◽  
Vol 2012 ◽  
pp. 1-10
Author(s):  
Ruchi Das ◽  
Tarun Das
Keyword(s):  
Linear Space ◽  
Metric Space ◽  
Normed Linear Space ◽  
Asymptotic Properties ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Bounded Subset ◽  
Necessary And Sufficient

We define and study the notions of positively and negatively -asymptotic points for a homeomorphism on a metric -space. We obtain necessary and sufficient conditions for two points to be positively/negatively -asymptotic. Also, we show that the problem of studying -expansive homeomorphisms on a bounded subset of a normed linear -space is equivalent to the problem of studying linear -expansive homeomorphisms on a bounded subset of another normed linear -space.

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