scholarly journals Volterra spaces revisited

2005 ◽  
Vol 79 (1) ◽  
pp. 61-76 ◽  
Author(s):  
Jiling Cao ◽  
David Gauld
Keyword(s):  
Homogeneous Space ◽  
Vietoris Topology ◽  
Topological Properties ◽  
Category Theorem ◽  
Nonempty Compact ◽  
Compact Subsets

AbstractIn this paper, we investigate Volterra spaces and relevant topological properties. New characterizations of weakly Volterra spaces are provided. An analogy of the Banach category theorem in terms of Volterra properties is obtained. It is shown that every weakly Volterra homogeneous space is Volterra, and there are metrizable Baire spaces whose hyperspaces of nonempty compact subsets endowed with the Vietoris topology are not weakly Volterra.

scholarly journals Topologies on Superspaces of TVS-Cone Metric Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-5 ◽  
Author(s):  
Xun Ge ◽  
Shou Lin
Keyword(s):  
Metric Space ◽  
Metric Spaces ◽  
Cone Metric Space ◽  
Vietoris Topology ◽  
Cone Metric Spaces ◽  
Nonempty Compact ◽  
Cone Metric ◽  
Compact Subsets

This paper investigates superspaces𝒫0(X)and𝒦0(X)of a tvs-cone metric space(X,d), where𝒫0(X)and𝒦0(X)are the space consisting of nonempty subsets ofXand the space consisting of nonempty compact subsets ofX, respectively. The purpose of this paper is to establish some relationships between the lower topology and the lower tvs-cone hemimetric topology (resp., the upper topology and the upper tvs-cone hemimetric topology to the Vietoris topology and the Hausdorff tvs-cone hemimetric topology) on𝒫0(X)and𝒦0(X), which makes it possible to generalize some results of superspaces from metric spaces to tvs-cone metric spaces.

Topological properties of the multifunction space L(X) of cusco maps

2008 ◽  
Vol 58 (6) ◽  
Author(s):  
Ľ. Holá ◽  
Tanvi Jain ◽  
R. McCoy
Keyword(s):  
Topological Space ◽  
Real Line ◽  
Vietoris Topology ◽  
Topological Properties ◽  
Connected Subset ◽  
The Real ◽  
Upper Semicontinuous ◽  
Set Valued Mapping ◽  
Nonempty Compact

AbstractA set-valued mapping F from a topological space X to a topological space Y is called a cusco map if F is upper semicontinuous and F(x) is a nonempty, compact and connected subset of Y for each x ∈ X. We denote by L(X), the space of all subsets F of X × ℝ such that F is the graph of a cusco map from the space X to the real line ℝ. In this paper, we study topological properties of L(X) endowed with the Vietoris topology.

scholarly journals Vietoris topology on spaces dominated by second countable ones

2015 ◽  
Vol 13 (1) ◽  
Author(s):  
Carlos Islas ◽  
Daniel Jardon
Keyword(s):  
Compact Space ◽  
Discrete Space ◽  
Vietoris Topology ◽  
The Family ◽  
Nonempty Compact ◽  
The One ◽  
Eberlein Compact ◽  
One Point Compactification ◽  
Compact Subsets

AbstractFor a given space X let C(X) be the family of all compact subsets of X. A space X is dominated by a space M if X has an M-ordered compact cover, this means that there exists a family F = {FK : K ∈ C(M)} ⊂ C(X) such that ∪ F = X and K ⊂ L implies that FK ⊂ FL for any K;L ∈ C(M). A space X is strongly dominated by a space M if there exists an M-ordered compact cover F such that for any compact K ⊂ X there is F ∈ F such that K ⊂ F . Let K(X) D C(X)\{Ø} be the set of all nonempty compact subsets of a space X endowed with the Vietoris topology. We prove that a space X is strongly dominated by a space M if and only if K(X) is strongly dominated by M and an example is given of a σ-compact space X such that K(X) is not Lindelöf†. It is stablished that if the weight of a scattered compact space X is not less than c, then the spaces Cp(K(X)) and K(Cp(X)) are not Lindelöf Σ. We show that if X is the one-point compactification of a discrete space, then the hyperspace K(X) is semi-Eberlein compact.

scholarly journals Li-Yorke Sensitivity of Set-Valued Discrete Systems

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Heng Liu ◽  
Fengchun Lei ◽  
Lidong Wang
Keyword(s):  
Metric Space ◽  
Short Proof ◽  
Hausdorff Metric ◽  
Discrete Systems ◽  
Compact Metric Space ◽  
Continuous Map ◽  
Nonempty Compact ◽  
Compact Subsets

Consider the surjective, continuous mapf:X→Xand the continuous mapf¯of𝒦(X)induced byf, whereXis a compact metric space and𝒦(X)is the space of all nonempty compact subsets ofXendowed with the Hausdorff metric. In this paper, we give a short proof that iff¯is Li-Yoke sensitive, thenfis Li-Yorke sensitive. Furthermore, we give an example showing that Li-Yorke sensitivity offdoes not imply Li-Yorke sensitivity off¯.

Domain representations of spaces of compact subsets

2010 ◽  
Vol 20 (2) ◽  
pp. 107-126 ◽  
Author(s):  
ULRICH BERGER ◽  
JENS BLANCK ◽  
PETTER KRISTIAN KØBER
Keyword(s):  
Metric Spaces ◽  
Topological Properties ◽  
Compact Sets ◽  
Natural Choice ◽  
Compact Subsets

We present a method for constructing from a given domain representation of a space X with underlying domain D, a domain representation of a subspace of compact subsets of X where the underlying domain is the Plotkin powerdomain of D. We show that this operation is functorial over a category of domain representations with a natural choice of morphisms. We study the topological properties of the space of representable compact sets and isolate conditions under which all compact subsets of X are representable. Special attention is paid to admissible representations and representations of metric spaces.

scholarly journals On star-K-Hurewicz spaces

Filomat ◽  
2017 ◽  
Vol 31 (5) ◽  
pp. 1279-1285 ◽  
Author(s):  
Yan-Kui Song
Keyword(s):  
Topological Properties ◽  
The Relationship ◽  
Compact Subsets

A space X is star-K-Hurewicz if for each sequence (Un : n ? N) of open covers of X there exists a sequence (Kn : n ? N) of compact subsets of X such that for each x ? X, x ? St(Kn,Un) for all but finitely many n. In this paper, we investigate the relationship between star-K-Hurewicz spaces and related spaces by giving some examples, and also study topological properties of star-K-Hurewicz spaces.

scholarly journals A note on compact sets in spaces of subsets

1988 ◽  
Vol 38 (3) ◽  
pp. 393-395 ◽  
Author(s):  
Phil Diamond ◽  
Peter Kloeden
Keyword(s):  
Metric Space ◽  
Complete Metric Space ◽  
Hausdorff Metric ◽  
Compact Sets ◽  
Selection Theorem ◽  
Starshaped Sets ◽  
Nonempty Compact ◽  
Compact Subsets

A simple characterisation is given of compact sets of the space K(X), of nonempty compact subsets of a complete metric space X, with the Hausdorff metric dH. It is used to give a new proof of the Blaschke selection theorem for compact starshaped sets.

scholarly journals Generalized modular fractal spaces and fixed point theorems

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Alireza Alihajimohammad ◽  
Reza Saadati
Keyword(s):  
Fixed Point ◽  
Hausdorff Distance ◽  
Fixed Point Theorems ◽  
Iterated Function System ◽  
Function System ◽  
Topological Properties ◽  
Iterated Function ◽  
Fractal Space ◽  
Nonempty Compact ◽  
Contraction Theory

AbstractIn this article, we introduce a new concept of Hausdorff distance through generalized modular metric on nonempty compact subsets and study some topological properties of it. This concept with contraction theory and the iterated function system (IFS) helps us to define a generalized modular fractal space.

scholarly journals Induced dynamics in hyperspaces of non-autonomous discrete systems

Filomat ◽  
2019 ◽  
Vol 33 (7) ◽  
pp. 1911-1920
Author(s):  
Radhika Vasisht ◽  
Ruchi Das
Keyword(s):  
Dynamical System ◽  
Autonomous System ◽  
Autonomous Systems ◽  
Discrete Systems ◽  
Vietoris Topology ◽  
Shadowing Property ◽  
Dynamical Properties ◽  
Autonomous Dynamical System ◽  
Compact Subsets

In this paper, the interrelations of some dynamical properties of a non-autonomous dynamical system (X, f1, ?) and its induced non-autonomous dynamical system (K(X), f1, ?) are studied, where K(X) is the hyperspace of all non-empty compact subsets of X, endowed with Vietoris topology. Various stronger forms of sensitivity and transitivity are considered. Some examples of non-autonomous systems are provided to support the results. A relation between shadowing property of the non-autonomous system (X, f1, ?) and its induced system (K(X), f1, ?) is studied.

scholarly journals An ideal version of the star-C-Hurewicz covering property

Filomat ◽  
2019 ◽  
Vol 33 (19) ◽  
pp. 6385-6393
Author(s):  
Sumit Singh ◽  
Brij Tyagi ◽  
Manoj Bhardwaj
Keyword(s):  
Hausdorff Space ◽  
Winning Strategy ◽  
Topological Properties ◽  
Covering Property ◽  
Countably Compact ◽  
Compact Subsets

A space X is said to have the star-C-I-Hurewicz (SCIH) property if for each sequence (Un : n ? N) of open covers of X there is a sequence (Kn : n ? N) of countably compact subsets of X such that for each x ? X, {n ? N : x ? St(Kn,Un)} ? I, where I is a proper admissible ideal of N. We investigate the relationships among the SCIH and related properties. We study the topological properties of the SCIH property. This paper generalizes several results of [21, 24] to the larger class of spaces having the SCIH property. The star-C-I-Hurewicz game is introduced. It is shown that, in a paracompact Hausdorff space X, if TWO has a winning strategy in the SCIH game on X, then TWO has a winning strategy in the I-Hurewicz game on X.

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