Voas, Charles H. · University of North Texas Digital Library (University of North Texas) · 1976
This paper is an exposition of the theory of the hyperspaces 2^X and C(X) of a topological space X. These spaces are obtained from X by collecting the nonempty closed and nonempty closed connected subsets respectively, and are topologized by the Vietoris topology. \nThe paper is organized in terms of increasing specialization of spaces, beginning with T1 spaces and proceeding through compact spaces, compact metric spaces and metric continua. Several basic techniques in hyperspace theory are discussed, and these techniques are applied to elucidate the topological structure of hyperspaces.
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The MIT Press eBooks · 1999
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Charles H. Voas · University of North Texas Digital Library (University of North Texas) · 1976
This paper is an exposition of the theory of the hyperspaces 2^X and C(X) of a topological space X. These spaces are obtained from X by collecting the nonempty closed and nonempty closed connected subsets respectively, and are topologized by the Vietoris topology. The paper is organized in terms of increasing specialization of spaces, beginning with T1 spaces and proceeding through compact spaces, compact metric spaces and metric continua. Several basic techniques in hyperspace theory are discussed, and these techniques are applied to elucidate the topological structure of hyperspaces.
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Anne Laure Vernet, Ami Skånberg · SPIRE - Sciences Po Institutional REpository · 2022
Performance artistique
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Verónica Martı́nez-de-la-Vega, Norberto Ordoñez · Topology and its Applications · 2012
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Abreu,Teresa, Corbacho,Eusébio, Tarieladze,Vaja · Portuguese National Funding Agency for Science, Research and Technology (RCAAP Project by FCT) · 2008
In this paper first we present elements of a theory of integration of a quasi-uniform conoid valued functions with respect to a positive measure following the doctoral thesis of the first author "Integration on Uniform Type Conoids". Then we show that it is possible to apply this theory for integration of functions with values in the hyperspace of non-empty convex subsets of a given quasi-uniform conoid. To realize such a possibility we treat the considered hyperspace as a quasi-uniform conoid in a rather natural way (as this is done in our joint work "Uniform type Hyperspaces").
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Mark Burgin · International Journal of Pure Mathematics · 2021
The development of mathematics brought mathematicians to infinite structures. This process started with transcendent real numbers and infinite sequences going through infinite series to transfinite numbers to nonstandard numbers to hypernumbers. From mathematics, infinity came to physics where physicists have been trying to get rid of infinity inventing a variety of techniques for doing this. In contrast to this, mathematicians as well as some physicists suggested ways to work with infinity introducing new mathematical structures such distributions and extrafunctions. The goal of this paper is to extend mathematical tools for treating infinity by considering hyperspaces and developing their theory.
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Patricia Pellicer-Covarrubias · Topology and its Applications · 2003
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Rodrigo Hernández-Gutiérrez, Alejandro Illanes, Verónica Martı́nez-de-la-Vega · Rocky Mountain Journal of Mathematics · 2015
Given a metric continuum $X$, we consider the following hyperspaces of $X$: $2^{X}$, $C_{n}(X)$ and $F_{n}(X)$ ($n\in \mathbb{N}$). Let $F_{1}(X)=\{\{x\}:x\in X\}$. A hyperspace $K(X)$ of $X$ is said to be rigid, provided that for every homeomorphism $h:K(X)\rightarrow K(X)$, we have $h(F_{1}(X))=F_{1}(X)$. In this paper, we study conditions under which a continuum $X$ has a rigid hyperspace $C_{n}(X)$. Among others, we consider families of continua, such as dendroids, Peano continua, hereditarily indecomposable continua and smooth fans.
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Alejandro Illanes · Elsevier eBooks · 2007
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Włodzimierz J. Charatonik, Matt Insall · Proceedings of the American Mathematical Society · 2016
From a measure space, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper X comma mu Superscript double-struck upper X Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>,</mml:mo> <mml:msup> <mml:mi> μ </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">X</mml:mi> </mml:mrow> </mml:mrow> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(X, \mu ^{\mathbb {X}})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> we define a measure <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mu Superscript double-struck upper P left-parenthesis double-struck upper X right-parenthesis"> <mml:semantics> <mml:msup> <mml:mi> μ </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">P</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">X</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">\mu ^{\mathbb {P}(\mathbb {X})}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on the power set of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . If <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper X comma tau right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>,</mml:mo> <mml:mi> τ </mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(X, \tau )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a compactum, whose topology <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="tau"> <mml:semantics> <mml:mi> τ </mml:mi> <mml:annotation encoding="application/x-tex">\tau</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is compatible with the measure <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mu Superscript double-struck upper X"> <mml:semantics> <mml:msup> <mml:mi> μ </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">X</mml:mi> </mml:mrow> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">\mu ^{\mathbb {X}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , then the measure <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mu Superscript double-struck upper P left-parenthesis double-struck upper X right-parenthesis"> <mml:semantics> <mml:msup> <mml:mi> μ </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">P</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">X</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">\mu ^{\mathbb {P}(\mathbb {X})}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> restricts to a natural measure on the hyperspace of closed sets of that given compactum. Surprisingly, under very mild conditions, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mu Superscript double-struck upper P left-parenthesis double-struck upper X right-parenthesis"> <mml:semantics> <mml:msup> <mml:mi> μ </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">P</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">X</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">\mu ^{\mathbb {P}(\mathbb {X})}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is always supported on the hyperspace of finite subsets.
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E. López-Pezoa, G. Mora, D. A. Redtwitz · Revista de la Real Academia de Ciencias Exactas Físicas y Naturales Serie A Matemáticas · 2024
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Alejandro Illanes · Universitext · 2025
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Alejandro Illanes, Paweł Krupski · Topology and its Applications · 2011
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A. Di Concilio · 2013
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Michael Hrušák, Fernando Hernández‐Hernández, Iván Martínez-Ruiz · Topology and its Applications · 2007
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Raúl Escobedo, María de J. López, Hugo Villanueva · Topology and its Applications · 2012
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Sergio Macı́as · 2018
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Jorge M. Martínez-Montejano · The Research Repository @ WVU (West Virginia University) · 2004
A characterization of Cp(X), the family of subcontinua of X containing a fixed point of X, when X is an atriodic continuum is given as follows. Assume Z is a continuum and consider the following three conditions: (1) Z is a planar absolute retract; (2) cut points of Z have component number two; (3) any true cyclic element of Z contains at most two cut points of Z. If X is an atriodic continuum and p ∈ X, then Cp(X) satisfies (1)--(3) and, conversely, if Z satisfies (1)--(3), then there exist an arc-like continuum (hence, atriodic) X and a point p ∈ X such that Cp(X) is homeomorphic to Z. For n ≥ 3, it is shown that the n th symmetric product of nondegenerate continua is mutually aposyndetic, and that the natural map of the Cartesian product onto the n th symmetric product of nondegenerate continua is not k-confluent for any positive integer k. It is also shown that for every nondegenerate continuum X there is a non-k-confluent map of some continuum onto F2(X) for any positive integer k. Answers are provided to questions of S. Macias and S. B. Nadler, Jr., about when the space of singletons is a Z-set in the hyperspace C( X). In answering one of these questions it is shown in general that C(X) being contractible is sufficient, but not necessary, for X to be a C(X)-coselection space.
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Nadler, Sam B.
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