Induced universal maps and some hyperspaces with the fixed point property

Nadler, Sam B.

doi:10.1090/s0002-9939-1987-0894449-4

Cited by 7 publications

(2 citation statements)

References 10 publications

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“…In this article we use this map to find a fixed point for its inducing map. The induced map has been used by Nadler in [12], to discuss the fixed point property of some hyperspaces of certain continua.…”

Section: Induced Map On Hyperspacesmentioning

confidence: 99%

Induced Homeomorphism and Atsuji Hyperspaces

Gupta¹,

Mukherjee²

2022

Preprint

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Given uniformly homeomorphic metric spaces X and Y , it is proved that the hyperspaces C(X) and C(Y ) are uniformly homeomorphic, where C(X) denotes the collection of all nonempty closed subsets of X, and is endowed with Hausdorff distance. Gerald Beer has proved that the hyperspace C(X) is Atsuji when X is either compact or uniformly discrete. An Atsuji space is a generalization of compact metric spaces as well as of uniformly discrete spaces. In this article, we investigate the space C(X) when X is Atsuji, and a class of Atsuji subspaces of C(X) is obtained. Using the obtained results, some fixed point results for continuous maps on Atsuji spaces are obtained. 2020 Mathematics Subject Classification. 54B20.

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Section: Induced Map On Hyperspacesmentioning

confidence: 99%

Induced Homeomorphism and Atsuji Hyperspaces

Gupta¹,

Mukherjee²

2022

Preprint

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“…The families 2 X and C(X) equipped with the Vietoris topology (see, e.g., [25, p. 10] for the definition) are called hyperspaces of X. The reader is referred to [15] and to [25] for needed information on hyperspaces.…”

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confidence: 99%

A Degree of Nonlocal Connectedness

Charatonik¹,

Charatonik²

2001

Rocky Mountain J. Math.

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To any continuum X we assign an ordinal number (or the symbol ∞) s(X), called the degree of nonlocal connectedness of X. We show that (1) the degree cannot be increased under continuous surjections; (2) for hereditarily unicoherent continua X, the degree of a subcontinuum of X is less than or equal to s(X); (3) s(C(X)) ≤ s(X), where C(X) denotes the hyperspace of subcontinua of a continuum X. We also investigate the degrees of Cartesian products and inverse limits. As an application we construct an uncountable family of metric continua X homeomorphic to C(X).Introduction. The idea of using ordinal numbers as a "measure" of some local or global properties of (compact) spaces is not new. Usually these properties are related to (non-)connectedness, and the defined "measure" can be used as a tool in studying various other properties of investigated spaces, both structural (internal) and mapping (external) ones. For example, Iliadis in [14] defines the notion of a normal sequence for hereditarily decomposable and hereditarily unicoherent metric continua (i.e., for λ-dendroids) as follows. Let X be such a continuum. A continuum H ⊂ X is said to be in I(X) if, given any decomposition of X into finitely many subcontinua, H is contained in one element of the decomposition. Let Σ = {H α : α < λ} be a transfinite sequence of subcontinua of X, where λ is some countable ordinal number. Then Σ is called a normal sequence if (i) H 0 = X, (ii) H β = I(H α ) for ordinals β = α + 1, (iii) H β = ∩{H α : α < β} for limit ordinals β, and (iv) for each α < λ the continuum H α is nondegenerate. The least upper bound (or minimum) k(X) of the lengths of all normal sequences in X is called the depth of X. The concept was used to study various phenomena in λ-dendroids. For its modification, see [31].

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Induced Maps on n-Fold Hyperspaces

López

Macı́as

2018

Topics on Continua

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Induced universal maps and some hyperspaces with the fixed point property

Cited by 7 publications

References 10 publications

Induced Homeomorphism and Atsuji Hyperspaces

Induced Homeomorphism and Atsuji Hyperspaces

A Degree of Nonlocal Connectedness

Induced Maps on n-Fold Hyperspaces

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