On the hyperspace C n ( X ) / C n K ( X )

José G. Anaya; Enrique Castañeda-Alvarado; José A. Martínez-Cortez

Commentationes Mathematicae Universitatis Carolinae (2021)

  • Issue: 2, page 201-224
  • ISSN: 0010-2628

Abstract

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Let X be a continuum and n a positive integer. Let C n ( X ) be the hyperspace of all nonempty closed subsets of X with at most n components, endowed with the Hausdorff metric. For K compact subset of X , define the hyperspace C n K ( X ) = { A ∈ C n ( X ) : K ⊂ A } . In this paper, we consider the hyperspace C K n ( X ) = C n ( X ) / C n K ( X ) , which can be a tool to study the space C n ( X ) . We study this hyperspace in the class of finite graphs and in general, we prove some properties such as: aposyndesis, local connectedness, arcwise disconnectedness, and contractibility.

How to cite

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Anaya, José G., Castañeda-Alvarado, Enrique, and Martínez-Cortez, José A.. "On the hyperspace $C_n(X)/{C_n}_K(X)$." Commentationes Mathematicae Universitatis Carolinae (2021): 201-224. <http://eudml.org/doc/298106>.

@article{Anaya2021,
abstract = {Let $X$ be a continuum and $n$ a positive integer. Let $C_n(X)$ be the hyperspace of all nonempty closed subsets of $X$ with at most $n$ components, endowed with the Hausdorff metric. For $K$ compact subset of $X$, define the hyperspace $\{C_n\}_K(X)=\lbrace A\in C_n(X)\colon K\subset A\rbrace $. In this paper, we consider the hyperspace $C_K^n(X)=C_n(X)/\{C_n\}_K(X)$, which can be a tool to study the space $C_n(X)$. We study this hyperspace in the class of finite graphs and in general, we prove some properties such as: aposyndesis, local connectedness, arcwise disconnectedness, and contractibility.},
author = {Anaya, José G., Castañeda-Alvarado, Enrique, Martínez-Cortez, José A.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {hyperspace; continuum; containment hyperspace; aposyndesis; finite graph; Peano continuum; contractibility},
language = {eng},
number = {2},
pages = {201-224},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {On the hyperspace $C_n(X)/\{C_n\}_K(X)$},
url = {http://eudml.org/doc/298106},
year = {2021},
}

TY - JOUR
AU - Anaya, José G.
AU - Castañeda-Alvarado, Enrique
AU - Martínez-Cortez, José A.
TI - On the hyperspace $C_n(X)/{C_n}_K(X)$
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2021
PB - Charles University in Prague, Faculty of Mathematics and Physics
IS - 2
SP - 201
EP - 224
AB - Let $X$ be a continuum and $n$ a positive integer. Let $C_n(X)$ be the hyperspace of all nonempty closed subsets of $X$ with at most $n$ components, endowed with the Hausdorff metric. For $K$ compact subset of $X$, define the hyperspace ${C_n}_K(X)=\lbrace A\in C_n(X)\colon K\subset A\rbrace $. In this paper, we consider the hyperspace $C_K^n(X)=C_n(X)/{C_n}_K(X)$, which can be a tool to study the space $C_n(X)$. We study this hyperspace in the class of finite graphs and in general, we prove some properties such as: aposyndesis, local connectedness, arcwise disconnectedness, and contractibility.
LA - eng
KW - hyperspace; continuum; containment hyperspace; aposyndesis; finite graph; Peano continuum; contractibility
UR - http://eudml.org/doc/298106
ER -

References

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  1. Anaya J. G., Castañeda-Alvarado E., Fuentes-Montes de Oca A., Orozco-Zitli F., Making holes in the cone, suspension and hyperspaces of some continua, Comment. Math. Univ. Carolin. 59 (2018), no. 3, 343–364. MR3861557
  2. Baik B. S., Hur K., Rhee C. J., R i -sets and contractibility, J. Korean Math. Soc. 34 (1997), no. 2, 309–319. MR1455544
  3. Bennett D. E., 10.2140/pjm.1971.37.585, Pacific J. Math. 37 (1971), 585–589. MR0305370DOI10.2140/pjm.1971.37.585
  4. Camargo J., Macías S., Quotients of n -fold hyperspaces, Topology Appl. 197 (2016), 154–166. MR3426913
  5. Castañeda-Alvarado E., Mondragón R. C., Ordoñez N., Orozco-Zitli F., The hyperspace F n K ( X ) , Bull. Iranian Math. Soc. 47 (2021), no. 3, 659–678. MR4249170
  6. Charatonik J. J., 10.1016/S0166-8641(89)80009-4, Topology Appl. 33 (1989), no. 2, 209–215. MR1020282DOI10.1016/S0166-8641(89)80009-4
  7. Charatonik J. J., Recent research in hyperspace theory, Extracta Math. 18 (2003), no. 2, 235–262. MR2002449
  8. Duda R., 10.4064/fm-62-3-265-286, Fund. Math. 62 (1968), 265–286. MR0236881DOI10.4064/fm-62-3-265-286
  9. Escobedo R., López M. de J., Macías S., 10.1016/j.topol.2003.08.024, Topology Appl. 138 (2004), no. 1–3, 109–124. MR2035475DOI10.1016/j.topol.2003.08.024
  10. Hernández-Gutiérrez R., Martínez-de-la-Vega V., 10.1016/j.topol.2013.06.001, Topology Appl. 160 (2013), no. 13, 1577–1587. MR3091331DOI10.1016/j.topol.2013.06.001
  11. Illanes Mejía A., Hiperespacios de continuos, Aportaciones Matemáticas, Serie Textos, 28, Sociedad Matemática Mexicana, México, 2004 (Spanish). MR2111741
  12. Illanes A., Nadler S. B., Jr., Hyperspaces, Fundamentals and Recent Advances, Monographs and Textbooks in Pure and Applied Mathematics, 216, Marcel Dekker, New York, 1999. MR1670250
  13. Macías J. C., 10.3336/gm.43.2.14, Glas. Mat. Ser. III 43(63) (2008), 439–449. MR2460710DOI10.3336/gm.43.2.14
  14. Macías S., 10.1090/S0002-9939-97-04175-0, Proc. Amer. Math. Soc. 125 (1997), no. 10, 3069–3073. MR1425134DOI10.1090/S0002-9939-97-04175-0
  15. Macías S., On the hyperspaces 𝒞 n ( X ) of a continuum X . II, Proc. of the 2000 Topology and Dynamics Conf., San Antonio, Topology Proc. 25 (2000), 255–276. MR1875596
  16. Macías S., 10.1016/S0166-8641(99)00151-0, Topology Appl. 109 (2001), no. 2, 237–256. MR1806337DOI10.1016/S0166-8641(99)00151-0
  17. Macías S., 10.1016/j.topol.2003.08.023, Topology Appl. 138 (2004), no. 1–3, 125–138. MR2035476DOI10.1016/j.topol.2003.08.023
  18. Macías S., On the n -fold hyperspace suspension of continua. II, Glas. Mat. Ser. III 41(61) (2006), no. 2, 335–343. MR2282743
  19. Martínez-de-la-Vega V., Dimension of n -fold hyperspaces of graphs, Houston J. Math. 32 (2006), no. 3, 783–799. MR2247910
  20. Nadler S. B., Jr., Hyperspaces of Sets, Monographs and Textbooks in Pure and Applied Mathematics, 49, Marcel Dekker, New York, 1978. Zbl1125.54001MR0500811
  21. Nadler S. B., Jr., A fixed point theorem for hyperspaces suspensions, Houston J. Math. 5 (1979), no. 1, 125–132. MR0533646
  22. Nadler S. B., Jr., Continuum Theory, An Introduction, Monographs and Textbooks in Pure and Applied Mathematics, 158, Marcel Dekker, New York, 1992. Zbl0819.54015MR1192552
  23. Nadler S. B., Jr., Dimension Theory, An Introduction with Exercises, Aportaciones Matemáticas, Serie Textos, 18, Sociedad Matemática Mexicana, México, 2002. MR1925171

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