Compact Sets in CP(X) and Calibers

1992 ◽  
Vol 35 (4) ◽  
pp. 497-502 ◽  
Author(s):  
N. D. Kalamidas ◽  
G. D. Spiliopoulos
Keyword(s):  
Compact Sets ◽  
Chain Conditions ◽  
Corson Compact ◽  
Compact Subsets

AbstractThis presentation concerns the relation of chain conditions on a space X, with the weights of compact sets in Cp(X), generalizing up to the class of dσ-bounded spaces, or stable spaces. In the last case, stronger results are obtained for Corson compact subsets of CP(X).

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.

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.

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.

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.

Gδ Sets IN σ-IDEALS GENERATED BY COMPACT SETS

2019 ◽  
Vol 84 (02) ◽  
pp. 781-797
Author(s):  
MAYA SARAN
Keyword(s):  
Compact Subset ◽  
Polish Space ◽  
Compact Sets ◽  
Natural Class ◽  
Image Position ◽  
Compact Subsets

AbstractGiven a compact Polish space E and the hyperspace of its compact subsets ${\cal K}\left( E \right)$, we consider Gδ σ-ideals of compact subsets of E. Solecki has shown that any σ-ideal in a broad natural class of Gδ ideals can be represented via a compact subset of ${\cal K}\left( E \right)$; in this article we examine the behaviour of Gδ subsets of E with respect to the representing set. Given an ideal I in this class, we construct a representing set that recognises a compact subset of E as being “small” precisely when it is in I, and recognises a Gδ subset of E as being “small” precisely when it is covered by countably many compact sets from I.

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 $$\omega ^\omega $$-Base and infinite-dimensional compact sets in locally convex spaces

2021 ◽  
Author(s):  
Taras Banakh ◽  
Jerzy Ka̧kol ◽  
Johannes Philipp Schürz
Keyword(s):  
Convex Space ◽  
Locally Convex Space ◽  
Compact Set ◽  
Compact Sets ◽  
Infinite Dimensional ◽  
Neighborhood Base ◽  
Remarkable Result ◽  
Locally Convex Topology ◽  
Locally Convex ◽  
Compact Subsets

AbstractA locally convex space (lcs) E is said to have an $$\omega ^{\omega }$$ ω ω -base if E has a neighborhood base $$\{U_{\alpha }:\alpha \in \omega ^\omega \}$$ { U α : α ∈ ω ω } at zero such that $$U_{\beta }\subseteq U_{\alpha }$$ U β ⊆ U α for all $$\alpha \le \beta $$ α ≤ β . The class of lcs with an $$\omega ^{\omega }$$ ω ω -base is large, among others contains all (LM)-spaces (hence (LF)-spaces), strong duals of distinguished Fréchet lcs (hence spaces of distributions $$D^{\prime }(\Omega )$$ D ′ ( Ω ) ). A remarkable result of Cascales-Orihuela states that every compact set in an lcs with an $$\omega ^{\omega }$$ ω ω -base is metrizable. Our main result shows that every uncountable-dimensional lcs with an $$\omega ^{\omega }$$ ω ω -base contains an infinite-dimensional metrizable compact subset. On the other hand, the countable-dimensional vector space $$\varphi $$ φ endowed with the finest locally convex topology has an $$\omega ^\omega $$ ω ω -base but contains no infinite-dimensional compact subsets. It turns out that $$\varphi $$ φ is a unique infinite-dimensional locally convex space which is a $$k_{\mathbb {R}}$$ k R -space containing no infinite-dimensional compact subsets. Applications to spaces $$C_{p}(X)$$ C p ( X ) are provided.

scholarly journals Necessary and sufficient conditions for precompact sets to be metrisable

2006 ◽  
Vol 74 (1) ◽  
pp. 7-13 ◽  
Author(s):  
J.C. Ferrando ◽  
J. Kasakol ◽  
M. López Pellicer
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Locally Convex Spaces ◽  
Compact Sets ◽  
Necessary And Sufficient ◽  
Locally Convex ◽  
Convex Spaces ◽  
Compact Subsets

This self-contained paper characterises those locally convex spaces whose (weakly) precompact (respectively, compact) subsets are metrisable. Applications and examples are provided. Our approach also applies to get Cascales-Orihuela's, Valdivia's and Robertson's metrisation theorems for (pre)compact sets.

scholarly journals Homeomorphisms of Compact Sets in Certain Hausdorff Spaces

2011 ◽  
Vol 2011 ◽  
pp. 1-11
Author(s):  
Arthur D. Grainger
Keyword(s):  
Compact Sets ◽  
Hausdorff Spaces ◽  
Nonempty Compact ◽  
Compact Subsets

We construct a class of Hausdorff spaces (compact and noncompact) with the property that nonempty compact subsets of these spaces that have the same cardinality are homeomorphic. Also, it is shown that these spaces contain compact subsets that are infinite.

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