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.

scholarly journals On the Fuzzy Number Space with the Level Convergence Topology

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
J. J. Font ◽  
A. Miralles ◽  
M. Sanchis
Keyword(s):  
Topological Space ◽  
Fuzzy Number ◽  
Continuous Functions ◽  
Topological Properties ◽  
Compact Sets ◽  
Compact Open Topology ◽  
Locally Compact ◽  
Compact Topological Space ◽  
Convergence Topology

We characterize compact sets of𝔼1endowed with the level convergence topologyτℓ. We also describe the completion(𝔼1̂,𝒰̂)of𝔼1with respect to its natural uniformity, that is, the pointwise uniformity𝒰, and show other topological properties of𝔼1̂, as separability. We apply these results to give an Arzela-Ascoli theorem for the space of(𝔼1,τℓ)-valued continuous functions on a locally compact topological space equipped with the compact-open topology.

A characterization of compact subsets of fuzzy number space

2001 ◽  
Vol 123 (2) ◽  
pp. 191-195 ◽  
Author(s):  
Byung Moon Ghil ◽  
Sang Yeol Joo ◽  
Yun Kyong Kim
Keyword(s):  
Fuzzy Number ◽  
Compact Subsets

A characterization of certain compact sets of measure zero

1975 ◽  
Vol 78 (2) ◽  
pp. 317-319
Author(s):  
D. M. Connolly ◽  
J. H. Williamson
Keyword(s):  
Measure Zero ◽  
Compact Sets ◽  
Single Generator ◽  
Compact Subsets

AbstractWe show that the only compact subsets of [0, 1] of measure zero are the obvious ones, and extend this idea to show that the only compact subsets of [0, 1] that generate semigroups of ℝ of measure zero are the obvious ones. We apply this last result to prove that any proper Raikov system of subsets of ℝ, with a single generator, is contained in a strictly larger proper Raikov system.

CHARACTERIZATIONS OF THE BOREL -FIELDS OF THE FUZZY NUMBER SPACE

2017 ◽  
Vol 58 (3-4) ◽  
pp. 265-275
Author(s):  
TAI-HE FAN ◽  
MENG-KE BIAN
Keyword(s):  
Approximation Method ◽  
Fuzzy Number ◽  
Fuzzy Numbers ◽  
Hausdorff Metric ◽  
Measurable Space ◽  
Borel Field ◽  
Applied Fields ◽  
Supremum Metric

In this paper, we characterize Borel $\unicode[STIX]{x1D70E}$-fields of the set of all fuzzy numbers endowed with different metrics. The main result is that the Borel $\unicode[STIX]{x1D70E}$-fields with respect to all known separable metrics are identical. This Borel field is the Borel $\unicode[STIX]{x1D70E}$-field making all level cut functions of fuzzy mappings from any measurable space to the fuzzy number space measurable with respect to the Hausdorff metric on the cut sets. The relation between the Borel $\unicode[STIX]{x1D70E}$-field with respect to the supremum metric $d_{\infty }$ is also demonstrated. We prove that the Borel field is induced by a separable and complete metric. A global characterization of measurability of fuzzy-valued functions is given via the main result. Applications to fuzzy-valued integrals are given, and an approximation method is presented for integrals of fuzzy-valued functions. Finally, an example is given to illustrate the applications of these results in economics. This example shows that the results in this paper are basic to the theory of fuzzy-valued functions, such as the fuzzy version of Lebesgue-like integrals of fuzzy-valued functions, and are useful in applied fields.

A Characterization of Compact Subsets of E1

1972 ◽  
Vol 79 (3) ◽  
pp. 278-279
Author(s):  
R. K. Tamaki
Keyword(s):  
Compact Subsets

Pego everywhere

2016 ◽  
Vol 15 (04) ◽  
pp. 1650074 ◽  
Author(s):  
Przemysław Górka ◽  
Tomasz Kostrzewa
Keyword(s):  
Fourier Transform ◽  
Abelian Groups ◽  
General Version ◽  
Pontryagin Duality ◽  
Locally Compact ◽  
Locally Compact Abelian Groups ◽  
Compact Abelian Groups ◽  
The Fourier Transform ◽  
Compact Subsets

In this note we show the general version of Pego’s theorem on locally compact abelian groups. The proof relies on the Pontryagin duality as well as on the Arzela–Ascoli theorem. As a byproduct, we get the characterization of relatively compact subsets of [Formula: see text] in terms of the Fourier transform.

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.

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