scholarly journals Paths in hyperspaces

2003 ◽  
Vol 4 (2) ◽  
pp. 377 ◽  
Author(s):  
Camillo Constantini ◽  
Wieslaw Kubís
Keyword(s):  
Metric Space ◽  
Metric Spaces ◽  
Hausdorff Metric ◽  
Absolute Retract ◽  
Almost Convex ◽  
Convex Metric Spaces ◽  
Metric Topology ◽  
Hausdorff Metric Topology ◽  
Bounded Sets ◽  
Separable Metric Space

<p>We prove that the hyperspace of closed bounded sets with the Hausdor_ topology, over an almost convex metric space, is an absolute retract. Dense subspaces of normed linear spaces are examples of, not necessarily connected, almost convex metric spaces. We give some necessary conditions for the path-wise connectedness of the Hausdorff metric topology on closed bounded sets. Finally, we describe properties of a separable metric space, under which its hyperspace with the Wijsman topology is path-wise connected.</p>

scholarly journals A note on suns in convex metric spaces

2010 ◽  
Vol 87 (101) ◽  
pp. 139-142
Author(s):  
T.D. Narang ◽  
R. Sangeeta
Keyword(s):  
Metric Space ◽  
Metric Spaces ◽  
Metric Projection ◽  
Convex Metric Space ◽  
Chebyshev Set ◽  
Almost Convex ◽  
Convex Metric Spaces ◽  
Lower Semi ◽  
Existence Set

We prove that in a convex metric space (X,d), an existence set K having a lower semi continuous metric projection is a ?-sun and in a complete M-space, a Chebyshev set K with a continuous metric projection is a ?-sun as well as almost convex.

Convergence of the Ishikawa Iterates for Multi-Valued Mappings in Convex Metric Spaces

2008 ◽  
Vol 15 (1) ◽  
pp. 39-43
Author(s):  
Ljubomir B. Ćirić ◽  
Nebojša T. Nikolić
Keyword(s):  
Metric Space ◽  
Convex Subset ◽  
Metric Spaces ◽  
Satisfy Condition ◽  
Convex Metric Space ◽  
Convex Metric Spaces ◽  
The Family

Abstract Let (𝑋, 𝑑) be a convex metric space, 𝐶 be a closed and convex subset of 𝑋 and let 𝐵(𝐶) be the family of all nonempty bounded subsets of 𝐶. In this paper some results are obtained on the convergence of the Ishikawa iterates associated with a pair of multi-valued mappings 𝑆,𝑇 : 𝐶 → 𝐵(𝐶) which satisfy condition (2.1) below.

scholarly journals A Universal Separable Diversity

2017 ◽  
Vol 5 (1) ◽  
pp. 138-151 ◽  
Author(s):  
David Bryant ◽  
André Nies ◽  
Paul Tupper
Keyword(s):  
Metric Space ◽  
Metric Spaces ◽  
Complete Metric Space ◽  
Urysohn Space ◽  
Fundamental Properties ◽  
Finite Set ◽  
Whole Space ◽  
Set Of Points ◽  
Separable Metric Space

AbstractThe Urysohn space is a separable complete metric space with two fundamental properties: (a) universality: every separable metric space can be isometrically embedded in it; (b) ultrahomogeneity: every finite isometry between two finite subspaces can be extended to an auto-isometry of the whole space. The Urysohn space is uniquely determined up to isometry within separable metric spaces by these two properties. We introduce an analogue of the Urysohn space for diversities, a recently developed variant of the concept of a metric space. In a diversity any finite set of points is assigned a non-negative value, extending the notion of a metric which only applies to unordered pairs of points.We construct the unique separable complete diversity that it is ultrahomogeneous and universal with respect to separable diversities.

scholarly journals Metric spaces with nice closed balls and distance functions for closed sets

1987 ◽  
Vol 35 (1) ◽  
pp. 81-96 ◽  
Author(s):  
Gerald Beer
Keyword(s):  
Distance Function ◽  
Metric Space ◽  
Pointwise Convergence ◽  
Metric Spaces ◽  
Closed Ball ◽  
Distance Functions ◽  
Entire Space ◽  
Closed Sets ◽  
Bounded Sets

A metric space 〈X,d〉 is said to have nice closed balls if each closed ball in X is either compact or the entire space. This class of spaces includes the metric spaces in which closed and bounded sets are compact and those for which the distance function is the zero-one metric. We show that these are the spaces in which the relation F = Lim Fn for sequences of closed sets is equivalent to the pointwise convergence of 〈d (.,Fn)〉 to d (.,F). We also reconcile these modes of convergence with three other closely related ones.

Weak-open images of locally separable metric spaces

2011 ◽  
Vol 48 (2) ◽  
pp. 145-159
Author(s):  
Zhaowen Li ◽  
Xun Ge ◽  
Qingguo Li
Keyword(s):  
Metric Space ◽  
Metric Spaces ◽  
Weak Base ◽  
Cosmic Space ◽  
Locally Separable Metric Spaces ◽  
Separable Metric Space

In this paper, we prove that a space X is a weak-open compact image of a locally separable metric space if and only if X has a uniform cosmic-weak-base if and only if X is a weak-open compact image of a metric space and a locally cosmic space, and give some internal characterizations of weak-open s-images of locally separable metric spaces.

scholarly journals On Mosco convergence of convex sets

1988 ◽  
Vol 38 (2) ◽  
pp. 239-253 ◽  
Author(s):  
Gerald Beer
Keyword(s):  
Convex Sets ◽  
Hausdorff Metric ◽  
Mosco Convergence ◽  
Natural Topology ◽  
Weakly Compact ◽  
Reflexive Space ◽  
Convergence Of Sequences ◽  
Metric Topology ◽  
Hausdorff Metric Topology

We present a natural topology compatible with the Mosco convergence of sequences of closed convex sets in a reflexive space, and characterise the topology in terms of the continuity of the distance between convex sets and fixed weakly compact ones. When the space is separable, the topology is Polish. As an application, we show that in this context, most closed convex sets are almost Chebyshev, a result that fails for the stronger Hausdorff metric topology.

scholarly journals Coarse infinite-dimensionality of hyperspaces of finite subsets

2021 ◽  
Author(s):  
Thomas Weighill ◽  
Takamitsu Yamauchi ◽  
Nicolò Zava
Keyword(s):  
Banach Space ◽  
Metric Space ◽  
Metric Spaces ◽  
Hausdorff Metric ◽  
Convex Banach Space ◽  
Uniformly Convex Banach Space ◽  
Uniformly Convex ◽  
Dimensional Properties ◽  
Bounded Geometry ◽  
Infinite Dimensional

AbstractWe consider infinite-dimensional properties in coarse geometry for hyperspaces consisting of finite subsets of metric spaces with the Hausdorff metric. We see that several infinite-dimensional properties are preserved by taking the hyperspace of subsets with at most n points. On the other hand, we prove that, if a metric space contains a sequence of long intervals coarsely, then its hyperspace of finite subsets is not coarsely embeddable into any uniformly convex Banach space. As a corollary, the hyperspace of finite subsets of the real line is not coarsely embeddable into any uniformly convex Banach space. It is also shown that every (not necessarily bounded geometry) metric space with straight finite decomposition complexity has metric sparsification property.

scholarly journals Coincidence best proximity points in convex metric spaces

Filomat ◽  
2018 ◽  
Vol 32 (7) ◽  
pp. 2451-2463 ◽  
Author(s):  
Moosa Gabeleh ◽  
Olivier Otafudu ◽  
Naseer Shahzad
Keyword(s):  
Integral Equation ◽  
Metric Space ◽  
Metric Spaces ◽  
Best Proximity Point ◽  
Best Proximity Points ◽  
Convex Metric Spaces

Let T,S : A U B ? A U B be mappings such that T(A) ? B,T(B)? A and S(A) ? A,S(B)?B. Then the pair (T,S) of mappings defined on A[B is called cyclic-noncyclic pair, where A and B are two nonempty subsets of a metric space (X,d). A coincidence best proximity point p ? A U B for such a pair of mappings (T,S) is a point such that d(Sp,Tp) = dist(A,B). In this paper, we study the existence and convergence of coincidence best proximity points in the setting of convex metric spaces. We also present an application of one of our results to an integral equation.

scholarly journals Metric spaces related to Abelian groups

2021 ◽  
Vol 22 (1) ◽  
pp. 169
Author(s):  
Amir Veisi ◽  
Ali Delbaznasab
Keyword(s):  
Metric Space ◽  
Metric Spaces ◽  
Additive Group ◽  
Ordered Group ◽  
Infinite Cyclic Group ◽  
Ordered Groups ◽  
The Family ◽  
Nonempty Compact ◽  
Metric Topology ◽  
Induced Topology

<p>When working with a metric space, we are dealing with the additive group (R, +). Replacing (R, +) with an Abelian group (G, ∗), offers a new structure of a metric space. We call it a G-metric space and the induced topology is called the G-metric topology. In this paper, we are studying G-metric spaces based on L-groups (i.e., partially ordered groups which are lattices). Some results in G-metric spaces are obtained. The G-metric topology is defined which is further studied for its topological properties. We prove that if G is a densely ordered group or an infinite cyclic group, then every G-metric space is Hausdorff. It is shown that if G is a Dedekind-complete densely ordered group, (X, d) a G-metric space, A ⊆ X and d is bounded, then f : X → G with f(x) = d(x, A) := inf{d(x, a) : a ∈ A} is continuous and further x ∈ cl<sub>X</sub>A if and only if f(x) = e (the identity element in G). Moreover, we show that if G is a densely ordered group and further a closed subset of R, K(X) is the family of nonempty compact subsets of X, e &lt; g ∈ G and d is bounded, then d′ (A, B) &lt; g if and only if A ⊆ N<sub>d</sub>(B, g) and B ⊆ N<sub>d</sub>(A, g), where N<sub>d</sub>(A, g) = {x ∈ X : d(x, A) &lt; g}, d<sub>B</sub>(A) = sup{d(a, B) : a ∈ A} and d′ (A, B) = sup{d<sub>A</sub>(B), d<sub>B</sub>(A)}.</p>

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.

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