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Journal articles on the topic 'Topology. Hyperspace'

Author: Grafiati

Published: 4 June 2021

Last updated: 9 February 2022

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1

Curtis, D. W. "Application of a Selection Theorem to Hyperspace Contractibility." Canadian Journal of Mathematics 37, no. 4 (August 1, 1985): 747–59. http://dx.doi.org/10.4153/cjm-1985-040-7.

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For X a metric continuum, 2X denotes the hyper space of all nonempty subcompacta, with the topology induced by the Hausdorff metric H, and C(X) ⊂ 2X the hyperspace of subcontinua. These hyperspaces are continua, in fact are arcwise-connected, since there exist order arcs between each hyperspace element and the element X. They also have trivial shape, i.e., maps of the hyperspaces into ANRs are homotopic to constant maps. For a detailed discussion of these and other general hyperspace properties, we refer the reader to Nadler's monograph [4].The question of hyperspace contractibility was first considered by Wojdyslawski [8], who showed that 2X and C(X) are contractible if X is locally connected. Kelley [2] gave a more general condition (now called property K) which is sufficient, but not necessary, for hyperspace contractibility. The continuum X has property K if for every there exists δ > 0 such that, for every pair of points x, y with d(x, y) < δ and every subcontinuum M containing x, there exists a subcontinuum N containing y with .

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2

Constantini, Camillo, and Wieslaw Kubís. "Paths in hyperspaces." Applied General Topology 4, no. 2 (October 1, 2003): 377. http://dx.doi.org/10.4995/agt.2003.2040.

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<p>We prove that the hyperspace of closed bounded sets with the Hausdor_ topology, over an almost convex metric space, is an absolute retract. Dense subspaces of normed linear spaces are examples of, not necessarily connected, almost convex metric spaces. We give some necessary conditions for the path-wise connectedness of the Hausdorff metric topology on closed bounded sets. Finally, we describe properties of a separable metric space, under which its hyperspace with the Wijsman topology is path-wise connected.</p>

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3

Illanes, Alejandro, and Verónica Martı́nez-de-la-Vega. "Product topology in the hyperspace of subcontinua." Topology and its Applications 105, no. 3 (August 2000): 305–17. http://dx.doi.org/10.1016/s0166-8641(99)00065-6.

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4

Holá, Lubica. "Embeddings in the Fell and Wijsman topologies." Filomat 33, no. 9 (2019): 2747–50. http://dx.doi.org/10.2298/fil1909747h.

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It is shown that if a T2 topological space X contains a closed uncountable discrete subspace, then the spaces (?1 + 1)? and (?1 + 1)?1 embed into (CL(X),?F), the hyperspace of nonempty closed subsets of X equipped with the Fell topology. If (X, d) is a non-separable perfect topological space, then (?1 + 1)? and (?1 +1)?1 embed into (CL(X), ?w(d)), the hyperspace of nonempty closed subsets of X equipped with the Wijsman topology, giving a partial answer to the Question 3.4 in [2].

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5

Naimpally, S. A., and P. L. Sharma. "Fine uniformity and the locally finite hyperspace topology." Proceedings of the American Mathematical Society 103, no. 2 (February 1, 1988): 641. http://dx.doi.org/10.1090/s0002-9939-1988-0943098-9.

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6

Hu, Thakyin, and Jen-Chun Fang. "Weak topology and Browder–Kirk's theorem on hyperspace." Journal of Mathematical Analysis and Applications 334, no. 2 (October 2007): 799–803. http://dx.doi.org/10.1016/j.jmaa.2006.12.078.

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7

Acosta, Gerardo. "Continua with almost unique hyperspace." Topology and its Applications 117, no. 2 (January 2002): 175–89. http://dx.doi.org/10.1016/s0166-8641(01)00018-9.

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8

Camargo, Javier, and Sergio Macías. "Embedding suspensions into hyperspace suspensions." Topology and its Applications 160, no. 10 (June 2013): 1115–22. http://dx.doi.org/10.1016/j.topol.2013.05.005.

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9

Vroegrijk, Tom. "Bornological modifications of hyperspace topologies." Topology and its Applications 161 (January 2014): 330–42. http://dx.doi.org/10.1016/j.topol.2013.10.035.

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10

García-Ferreira, S., and Y. F. Ortiz-Castillo. "The hyperspace of convergent sequences." Topology and its Applications 196 (December 2015): 795–804. http://dx.doi.org/10.1016/j.topol.2015.05.022.

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11

Shakir, Q. R. "On Vietoris Soft Topology I." Journal of Scientific Research 8, no. 1 (January 1, 2016): 13–19. http://dx.doi.org/10.3329/jsr.v8i1.23440.

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In this article, we define the hyperspace of soft closed sets of a soft topological space (FA, ?). In addition, we define the Vietoris soft topology, ?v, by determining the soft base of this topology which has the form ?FH1, FH2,.....FHn?, where ?FH1, FH2,.....FHn? are soft open sets in (FA, ?). Some properties of this topology are also investigated. The impact of introducing the Vietoris soft topology is to enable us to understand many properties of the structure of soft topologies corresponding to it.

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12

Maya, David, Patricia Pellicer-Covarrubias, and Roberto Pichardo-Mendoza. "Cardinal functions of the hyperspace of convergent sequences." Mathematica Slovaca 68, no. 2 (April 25, 2018): 431–50. http://dx.doi.org/10.1515/ms-2017-0114.

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Abstract The symbol 𝓢c(X) denotes the hyperspace of all nontrivial convergent sequences in a Hausdorff space X. This hyperspace is endowed with the Vietoris topology. In the current paper, we compare the cellularity, the tightness, the extent, the dispersion character, the net weight, the i-weight, the π-weight, the π-character, the pseudocharacter and the Lindelöf number of 𝓢c(X) with the corresponding cardinal function of X. We also answer a question posed by the authors in a previous paper.

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13

Naimpally, Somshekhar. "All hypertopologies are hit-and-miss." Applied General Topology 3, no. 1 (April 1, 2002): 45. http://dx.doi.org/10.4995/agt.2002.2111.

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<p>We solve a long standing problem by showing that all known hypertopologies are hit-and-miss. Our solution is not merely of theoretical importance. This representation is useful in the study of comparison of the Hausdorff-Bourbaki or H-B uniform topologies and the Wijsman topologies among themselves and with others. Up to now some of these comparisons needed intricate manipulations. The H-B uniform topologies were the subject of intense activity in the 1960's in connection with the Isbell-Smith problem. We show that they are proximally locally finite topologies from which the solution to the above problem follows easily. It is known that the Wijsman topology on the hyperspace is the proximal ball (hit-and-miss) topology in”nice” metric spaces including the normed linear spaces. With the introduction of a new far-miss topology we show that the Wijsman topology is hit-and-miss for all metric spaces. From this follows a natural generalization of the Wijsman topology to the hyperspace of any T<sub>1</sub> space. Several existing results in the literature are easy consequences of our work.</p>

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14

Rosa, Marco, and Paolo Vitolo. "Comparability of lower Attouch-Wets topologies." Filomat 31, no. 5 (2017): 1435–40. http://dx.doi.org/10.2298/fil1705435r.

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Beer and Di Concilio [4] have given necessary and sufficient conditions for a two-sided Attouch-Wets topology to contain another on the hyperspace of non-empty closed subsets of a metrizable space as determined by metrics compatible with the topology. In the present paper, we characterize comparability of lower Attouch-Wets topologies as determined by compatible metrics.

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15

Bourquin, Steven, and Laszlo Zsilinszky. "Baire spaces and hyperspace topologies revisited." Applied General Topology 15, no. 1 (April 1, 2014): 85. http://dx.doi.org/10.4995/agt.2014.1897.

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16

Anaya, José G., Enrique Castañeda-Alvarado, and Alejandro Fuentes-Montes de Oca. "Making holes in the hyperspace suspension." Topology and its Applications 265 (September 2019): 106816. http://dx.doi.org/10.1016/j.topol.2019.106816.

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17

Capulín, Félix, Enrique Castañeda-Alvarado, Norberto Ordoñez, and Marco A. Ruiz. "The hyperspace of T-closed subcontinua." Topology and its Applications 275 (April 2020): 107154. http://dx.doi.org/10.1016/j.topol.2020.107154.

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18

Di Maio, G., R. Lowen, S. A. Naimpally, and M. Sioen. "Gap functionals, proximities and hyperspace compactification." Topology and its Applications 153, no. 5-6 (December 2005): 924–40. http://dx.doi.org/10.1016/j.topol.2005.01.021.

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19

Herrera-Carrasco, David, Alejandro Illanes, María de J. López, and Fernando Macías-Romero. "Dendrites with unique hyperspace C2(X)." Topology and its Applications 156, no. 3 (January 2009): 549–57. http://dx.doi.org/10.1016/j.topol.2008.08.007.

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20

Prajs, Janusz R. "The terminal hyperspace of homogeneous continua." Topology and its Applications 157, no. 3 (February 2010): 536–47. http://dx.doi.org/10.1016/j.topol.2009.10.011.

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21

Camargo, Javier, David Maya, and Luis Ortiz. "The hyperspace of nonblockers of F1(X)." Topology and its Applications 251 (January 2019): 70–81. http://dx.doi.org/10.1016/j.topol.2018.10.007.

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22

Simon Romero, Likin C. "Kelley continua and the hyperspace Λ(X)." Topology and its Applications 266 (October 2019): 106867. http://dx.doi.org/10.1016/j.topol.2019.106867.

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23

Escobedo, Raúl, Marı́a de Jesús López, and Sergio Macías. "On the hyperspace suspension of a continuum." Topology and its Applications 138, no. 1-3 (March 2004): 109–24. http://dx.doi.org/10.1016/j.topol.2003.08.024.

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24

Acosta, Gerardo, David Herrera-Carrasco, and Fernando Macías-Romero. "Local dendrites with unique hyperspace C(X)." Topology and its Applications 157, no. 13 (August 2010): 2069–85. http://dx.doi.org/10.1016/j.topol.2010.05.005.

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25

Chinen, Naotsugu, and Akira Koyama. "On the symmetric hyperspace of the circle." Topology and its Applications 157, no. 17 (November 2010): 2613–21. http://dx.doi.org/10.1016/j.topol.2010.07.012.

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26

Herrera-Carrasco, David, and Fernando Macías-Romero. "Local dendrites with unique n-fold hyperspace." Topology and its Applications 158, no. 2 (February 2011): 244–51. http://dx.doi.org/10.1016/j.topol.2010.11.004.

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27

Bell, M. "The hyperspace of a compact space, I." Topology and its Applications 72, no. 1 (August 1996): 39–46. http://dx.doi.org/10.1016/0166-8641(96)00012-0.

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28

Hu, Jennifer Shueh-Inn, and Thakyin Hu. "Krein-Milman's Extreme Point Theorem and Weak Topology on Hyperspace." Taiwanese Journal of Mathematics 20, no. 3 (May 2016): 629–38. http://dx.doi.org/10.11650/tjm.20.2016.6411.

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29

Bazilevich, L. E. "Topology of the hyperspace of convex bodies of constant width." Mathematical Notes 62, no. 6 (December 1997): 683–87. http://dx.doi.org/10.1007/bf02355455.

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30

Hou, Ji-Cheng, and Paolo Vitolo. "Fell topology on the hyperspace of a non-Hausdorff space." Ricerche di Matematica 57, no. 1 (June 6, 2008): 111–25. http://dx.doi.org/10.1007/s11587-008-0032-y.

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31

Illanes, Alejandro. "Contractibility of the hyperspace of sequences, harmonic fan." Topology and its Applications 277 (May 2020): 107167. http://dx.doi.org/10.1016/j.topol.2020.107167.

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32

Macı́as, Sergio. "On the n-fold hyperspace suspension of continua." Topology and its Applications 138, no. 1-3 (March 2004): 125–38. http://dx.doi.org/10.1016/j.topol.2003.08.023.

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33

Costantini, Camillo, Ľubica Holá, and Paolo Vitolo. "Tightness, character and related properties of hyperspace topologies." Topology and its Applications 142, no. 1-3 (July 2004): 245–92. http://dx.doi.org/10.1016/j.topol.2004.02.007.

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34

Pia̧tkiewicz, Leszek, and László Zsilinszky. "On (strong) α-favorability of the Wijsman hyperspace." Topology and its Applications 157, no. 16 (October 2010): 2555–61. http://dx.doi.org/10.1016/j.topol.2010.07.034.

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35

Guy, Jean-Sébastien. "Bourdieu in hyperspace: from social topology to the space of flows." International Review of Sociology 28, no. 3 (September 2, 2018): 510–23. http://dx.doi.org/10.1080/03906701.2018.1529074.

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36

Bella, A. "Some cardinality properties of a hyperspace with the locally finite topology." Proceedings of the American Mathematical Society 104, no. 4 (April 1, 1988): 1274. http://dx.doi.org/10.1090/s0002-9939-1988-0969059-1.

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37

Higueras-Montaño, Luisa F. "A hyperspace of convex bodies arising from tensor norms." Topology and its Applications 275 (April 2020): 107149. http://dx.doi.org/10.1016/j.topol.2020.107149.

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38

Ordoñez, Norberto, César Piceno, and Hugo Villanueva. "Closed orbits and the hyperspace of 12-homogeneous continua." Topology and its Applications 289 (February 2021): 107456. http://dx.doi.org/10.1016/j.topol.2020.107456.

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39

Escobedo, Raúl, Norberto Ordoñez, Rusell-Aarón Quiñones-Estrella, and Hugo Villanueva. "The hyperspace of connected boundary subcontinua of a continuum." Topology and its Applications 290 (March 2021): 107573. http://dx.doi.org/10.1016/j.topol.2020.107573.

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40

Dijkstra, Jan J., and Jan van Mill. "On the Group of Homeomorphisms of the Real Line That Map the Pseudoboundary Onto Itself." Canadian Journal of Mathematics 58, no. 3 (June 1, 2006): 529–47. http://dx.doi.org/10.4153/cjm-2006-022-8.

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AbstractIn this paper we primarily consider two natural subgroups of the autohomeomorphism group of the real line ℝ, endowed with the compact-open topology. First, we prove that the subgroup of homeomorphisms that map the set of rational numbers ℚ onto itself is homeomorphic to the infinite power of ℚ with the product topology. Secondly, the group consisting of homeomorphisms that map the pseudoboundary onto itself is shown to be homeomorphic to the hyperspace of nonempty compact subsets of ℚ with the Vietoris topology. We obtain similar results for the Cantor set but we also prove that these results do not extend to ℝn for n ≥ 2, by linking the groups in question with Erdős space.

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41

Di Caprio, Debora, and Stephen Watson. "Orderability and continuous selections for Wijsman and Vietoris hyperspaces." Applied General Topology 4, no. 2 (October 1, 2003): 361. http://dx.doi.org/10.4995/agt.2003.2039.

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<p>Bertacchi and Costantini obtained some conditions equivalent to the existence of continuous selections for the Wijsman hyperspace of ultrametric Polish spaces. We introduce a new class of hypertopologies, the macro-topologies. Both the Wijsman topology and the Vietoris topology belong to this class. We show that subject to natural conditions, the base space admits a closed order such that the minimum map is a continuous selection for every macro-topology. In the setting of Polish spaces, these conditions are substantially weaker than the ones given by Bertacchi and Costantini. In particular, we conclude that Polish spaces satisfying these conditions can be endowed with a compatible order and that the minimum function is a continuous selection for the Wijsman topology, just as it is for [0; 1]. This also solves a problem implicitely raised in Bertacchi and Costantini's paper.</p>

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42

Di Maio, Giuseppe, Enrico Meccariello, and Somashekhar Naimpally. "Bombay hypertopologies." Applied General Topology 4, no. 2 (October 1, 2003): 421. http://dx.doi.org/10.4995/agt.2003.2042.

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<p>Recently it was shown that, in a metric space, the upper Wijsman convergence can be topologized with the introduction of a new far-miss topology. The resulting Wijsman topology is a mixture of the ball topology and the proximal ball topology. It leads easily to the generalized or g-Wijsman topology on the hyperspace of any topological space with a compatible LO-proximity and a cobase (i.e. a family of closed subsets which is closed under finite unions and which contains all singletons). Further generalization involving a topological space with two compatible LO-proximities and a cobase results in a new hypertopology which we call the Bombay topology. The generalized locally finite Bombay topology includes the known hypertopologies as special cases and moreover it gives birth to many new hypertopologies. We show how it facilitates comparison of any two hypertopologies by proving one simple result of which most of the existing results are easy consequences.</p>

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43

Illanes, Alejandro. "Hereditarily indecomposable Hausdorff continua have unique hyperspaces 2X and Cn(X)." Publications de l'Institut Math?matique (Belgrade) 89, no. 103 (2011): 49–56. http://dx.doi.org/10.2298/pim1103049i.

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Let X be a Hausdorff continuum (a compact connected Hausdorff space). Let 2X (respectively, Cn(X)) denote the hyperspace of nonempty closed subsets of X (respectively, nonempty closed subsets of X with at most n components), with the Vietoris topology. We prove that if X is hereditarily indecomposable, Y is a Hausdorff continuum and 2X (respectively Cn(X)) is homeomorphic to 2Y (respectively, Cn(Y )), then X is homeomorphic to Y.

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44

Yang, Zhong Qiang, and Bao Can Zhang. "The hyperspace of the regions below continuous maps with the fell topology." Acta Mathematica Sinica, English Series 28, no. 1 (December 13, 2011): 57–66. http://dx.doi.org/10.1007/s10114-012-0030-6.

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45

Camargo, Javier, Félix Capulín, Enrique Castañeda-Alvarado, and David Maya. "Continua whose hyperspace of nonblockers of F1(X) is a continuum." Topology and its Applications 262 (August 2019): 30–40. http://dx.doi.org/10.1016/j.topol.2019.05.007.

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46

Santiago-Santos, Alicia, and Noé Trinidad Tapia-Bonilla. "Topological properties on n-fold pseudo-hyperspace suspension of a continuum." Topology and its Applications 270 (February 2020): 106956. http://dx.doi.org/10.1016/j.topol.2019.106956.

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47

Cao, Jiling, and A. H. Tomita. "The Wijsman hyperspace of a metric hereditarily Baire space is Baire." Topology and its Applications 157, no. 1 (January 2010): 145–51. http://dx.doi.org/10.1016/j.topol.2009.04.039.

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48

Kato, Hisao. "On the property of kelley in the hyperspace and Whitney continua." Topology and its Applications 30, no. 2 (November 1988): 165–74. http://dx.doi.org/10.1016/0166-8641(88)90015-6.

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49

Wu, Xinxing. "Chaos of Transformations Induced Onto the Space of Probability Measures." International Journal of Bifurcation and Chaos 26, no. 13 (December 15, 2016): 1650227. http://dx.doi.org/10.1142/s0218127416502278.

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For a dynamical system [Formula: see text], let [Formula: see text] be its induced dynamical system on the space of Borel probability measures with weak*-topology. It is proved that [Formula: see text] is [Formula: see text]-transitive (resp., exact, uniformly rigid) if and only if [Formula: see text] is weakly mixing and [Formula: see text]-transitive (resp., exact, uniformly rigid), where [Formula: see text] is an [Formula: see text]-vector of integers. Moreover, some analogous results are obtained for the hyperspace.

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50

Vasisht, Radhika, and Ruchi Das. "Induced dynamics in hyperspaces of non-autonomous discrete systems." Filomat 33, no. 7 (2019): 1911–20. http://dx.doi.org/10.2298/fil1907911v.

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In this paper, the interrelations of some dynamical properties of a non-autonomous dynamical system (X, f1, ?) and its induced non-autonomous dynamical system (K(X), f1, ?) are studied, where K(X) is the hyperspace of all non-empty compact subsets of X, endowed with Vietoris topology. Various stronger forms of sensitivity and transitivity are considered. Some examples of non-autonomous systems are provided to support the results. A relation between shadowing property of the non-autonomous system (X, f1, ?) and its induced system (K(X), f1, ?) is studied.

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