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 Strong Li–Yorke Chaos for Time-Varying Discrete Dynamical Systems with A-Coupled-Expansion

2015 ◽  
Vol 25 (13) ◽  
pp. 1550186 ◽  
Author(s):  
Hua Shao ◽  
Yuming Shi ◽  
Hao Zhu
Keyword(s):  
Dynamical Systems ◽  
Metric Spaces ◽  
Discrete Systems ◽  
Discrete Dynamical Systems ◽  
Strong Sense ◽  
Time Varying ◽  
Transition Matrices ◽  
Complete Metric Spaces ◽  
Compact Subsets

This paper is concerned with strong Li–Yorke chaos induced by [Formula: see text]-coupled-expansion for time-varying (i.e. nonautonomous) discrete systems in metric spaces. Some criteria of chaos in the strong sense of Li–Yorke are established via strict coupled-expansions for irreducible transition matrices in bounded and closed subsets of complete metric spaces and in compact subsets of metric spaces, respectively, where their conditions are weaker than those in the existing literature. One example is provided for illustration.

scholarly journals A New Characterization of Compact Sets in Fuzzy Number Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Huan Huang ◽  
Congxin Wu
Keyword(s):  
Fuzzy Number ◽  
Compact Sets ◽  
Sequential Compactness ◽  
Convergence Topology ◽  
Compact Subsets

We give a new characterization of compact subsets of the fuzzy number space equipped with the level convergence topology. Based on this, it is shown that compactness is equivalent to sequential compactness on the fuzzy number space endowed with the level convergence topology. Our results imply that some previous compactness criteria are wrong. A counterexample also is given to validate this judgment.

Fixed Point Theorems for Maps With Local and Pointwise Contraction Properties

2018 ◽  
Vol 70 (3) ◽  
pp. 538-594 ◽  
Author(s):  
Krzysztof Chris Ciesielski ◽  
Jakub Jasinski
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorems ◽  
Metric Spaces ◽  
Topological Properties ◽  
Open Problems ◽  
Local Contraction ◽  
Path Connectedness ◽  
Perfect Subset ◽  
Minimal Dynamical System ◽  
Comprehensive Study

AbstractThis paper constitutes a comprehensive study of ten classes of self-maps on metric spaces ⟨X, d⟩ with the pointwise (i.e., local radial) and local contraction properties. Each of these classes appeared previously in the literature in the context of fixed point theorems.We begin with an overview of these fixed point results, including concise self contained sketches of their proofs. Then we proceed with a discussion of the relations among the ten classes of self-maps with domains ⟨X, d⟩ having various topological properties that often appear in the theory of fixed point theorems: completeness, compactness, (path) connectedness, rectifiable-path connectedness, and d-convexity. The bulk of the results presented in this part consists of examples of maps that show non-reversibility of the previously established inclusions between these classes. Among these examples, the most striking is a differentiable auto-homeomorphism f of a compact perfect subset X of ℝ with f′ ≡ 0, which constitutes also a minimal dynamical system. We finish by discussing a few remaining open problems on whether the maps with specific pointwise contraction properties must have the fixed points.

scholarly journals Isometries of spaces of unbounded continuous functions

2001 ◽  
Vol 63 (3) ◽  
pp. 475-484
Author(s):  
Jesús Araujo ◽  
Krzysztof Jarosz
Keyword(s):  
Banach Spaces ◽  
Metric Spaces ◽  
Continuous Functions ◽  
Topological Spaces ◽  
Compact Sets ◽  
Surjective Isometry ◽  
Bounded Continuous Functions ◽  
Stone Theorem

By the classical Banach-Stone Theorem any surjective isometry between Banach spaces of bounded continuous functions defined on compact sets is given by a homeomorphism of the domains. We prove that the same description applies to isometries of metric spaces of unbounded continuous functions defined on non compact topological spaces.

scholarly journals A Characterization for Compact Sets in the Space of Fuzzy Star-Shaped Numbers withLpMetric

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Zhitao Zhao ◽  
Congxin Wu
Keyword(s):  
Compact Sets ◽  
Compact Subsets

By means of some auxiliary lemmas, we obtain a characterization of compact subsets in the space of all fuzzy star-shaped numbers withLpmetric for1≤p<∞. The result further completes and develops the previous characterization of compact subsets given by Wu and Zhao in 2008.

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.

On the existence and uniqueness of certain measures

1961 ◽  
Vol 262 (1309) ◽  
pp. 159-178
Keyword(s):  
Existence And Uniqueness ◽  
Convex Sets ◽  
Convex Polytopes ◽  
Sufficient Conditions ◽  
Base Space ◽  
Compact Sets ◽  
Unique Measure ◽  
Set Function ◽  
Compact Subsets

Suppose given a positive set-function μ ( F ) in a base space R defined on a base class F of compact sets F . In this paper we obtain conditions under which μ ( F ) determines a unique measure m ( E ) in R , finite on all compact subsets of R , and such that μ ( F ) lies between the measure of F and that of the interior of F for every set F ∈ F . We assume μ ( F ) to satisfy certain inequalities which are clearly necessary for our conclusions and show that if the class F is sufficiently big then every set-function μ ( F ) satisfying these conditions does determine such a unique measure m ( E ). Different sufficient conditions on F are given according as the sets F in ( a ) are convex polytopes, or have analytic boundaries, ( b ) have sectionally analytic boundaries, or ( c ) are general compact sets, and it is shown by examples that these conditions cannot be relaxed too much. Thus the conclusions under ( a ) no longer hold in the plane if we assume that the sets are starlike polygons or convex sets with sectionally analytic boundaries. Nor is it possible to replace the sets under ( b ) by closed Jordan domains.

Completeness of hyperspaces of compact subsets of quasi-metric spaces

2010 ◽  
Vol 127 (3) ◽  
pp. 260-272 ◽  
Author(s):  
M. Ali-Akbari ◽  
M. Pourmahdian
Keyword(s):  
Metric Spaces ◽  
Compact Subsets
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