When does the Fell topology on a hyperspace of closed sets coincide with the meet of the upper Kuratowski and the lower Vietoris topologies?

Nogura, Tsugunori; Shakhmatov, Dmitri

doi:10.1016/0166-8641(95)00098-4

Cited by 49 publications

(34 citation statements)

References 12 publications

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“…It is also known that metrizable consonant spaces are hereditarily Baire [4]. These results prompted Nogura and Shakhmatov to ask whether consonant metrizable spaces arě Cech-complete [25,Problem 11.4]. The question was resolved by Bouziad [6] by showing that the statement all analytic metrizable consonant spaces are completely metrizable is independent of usual axioms for set theory.…”

Section: Introductionmentioning

confidence: 94%

“…An interesting class of spaces was introduced by Dolecki, Greco and Lechicki in [11,12]: a space X was called consonant if on the hyperspace F(X ) the co-compact topology coincides with the upper Kuratowski topology. This class became of great interest especially after it was noticed by Nogura and Shakhmatov [25] that two other classical hyperspace topologies, the Fell and Kuratowski topologies, (automatically) coincide on F(X ) provided X is consonant.…”

Section: Introductionmentioning

confidence: 99%

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Consonance and Cantor set-selectors

Gutev

2012

Open Mathematics

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Section: Introductionmentioning

confidence: 94%

Section: Introductionmentioning

confidence: 99%

Consonance and Cantor set-selectors

Gutev

2012

Open Mathematics

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“…The hit-or-miss topology τ f (see [28,29]) (also known as H-topology [9], Fell topology [4,5,21], Choquet-Matheron topology [26], or weak Vietoris topology [32]) on F is generated by the subbase…”

Section: Preliminariesmentioning

confidence: 99%

F-sensitivity and (F_1, F_2)-sensitivity between dynamical systems and their induced hyperspace dynamical systems

Li¹,

Zhao²,

Wang³

et al. 2017

J. Nonlinear Sci. Appl.

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“…every closed subspace of a metrizable consonant metric space is a Baire space). These results prompted Nogura and Shakhmatov in [8,Problem 11.4] to ask whether consonant metrizable spaces are (Čech-)complete. This remained one of the central open problems in the theory of consonance.…”

mentioning

confidence: 99%

“…The class of consonant spaces was introduced by Dolecki, Greco and Lechicki in [2], [3] and was recently studied rather intensively. It was noticed by Nogura and Shakhmatov in [8] that two other classical topologies (automatically) coincide on the set of closed subsets of a consonant space, namely, the Fell topology and the Kuratowski topology. In [3], among other things, it is proved that everyČech-complete space is consonant.…”

mentioning

confidence: 99%

Consonance and topological completeness in analytic spaces

Bouziad¹

1999

Proc. Amer. Math. Soc.

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Abstract. We give a set-valued criterion for a topological space X to be consonant, i.e. the upper Kuratowski topology on the family of all closed subsets of X coincides with the co-compact topology. This characterization of consonance is then used to show that the statement "every analytic metrizable consonant space is complete" is independent of the usual axioms of set theory. This answers a question by Nogura and Shakhmatov. It is also proved that continuous open surjections defined on a consonant space are compact covering.A topological space X is said to be consonant if the co-compact topology on the set of all closed subsets of X coincides with the upper Kuratowski topology. The class of consonant spaces was introduced by Dolecki, Greco and Lechicki in [2], [3] and was recently studied rather intensively. It was noticed by Nogura and Shakhmatov in [8] that two other classical topologies (automatically) coincide on the set of closed subsets of a consonant space, namely, the Fell topology and the Kuratowski topology. In [3], among other things, it is proved that everyČech-complete space is consonant. It is also known that metrizable consonant spaces are hereditarily Baire [1] (i.e. every closed subspace of a metrizable consonant metric space is a Baire space). These results prompted Nogura and Shakhmatov in [8, Problem 11.4] to ask whether consonant metrizable spaces are (Čech-)complete. This remained one of the central open problems in the theory of consonance. For coanalytic metrizable spaces, this problem is entirely solved: A co-analytic metrizable space X is consonant if and only if X is Polish (see [1]). In this note we show that the statement "every analytic metrizable consonant space is complete" is independent of the usual axioms of set theory (Theorem 6). It is also proved that continuous open surjections defined on a consonant space are compact covering (Corollary 8), which gives a generalization of the classical theorem of Pasynkov stated forČech-complete spaces. Corollary 8, along with a construction of Michael [6], allows us to give a new and topological proof of the nonconsonance of the usual space of rationals.This note is based on a characterization of consonance in terms of a special property of lower semicontinuous set-valued maps (Theorem 2). Concerning topological completeness, an important and recent result of this kind, achieved by van Mill,

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When does the Fell topology on a hyperspace of closed sets coincide with the meet of the upper Kuratowski and the lower Vietoris topologies?

Cited by 49 publications

References 12 publications

Consonance and Cantor set-selectors

Consonance and Cantor set-selectors

F-sensitivity and (F_1, F_2)-sensitivity between dynamical systems and their induced hyperspace dynamical systems

Consonance and topological completeness in analytic spaces

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