Abstract
We give a brief overview of n-fold hyperspaces.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
G. Acosta, Continua with Unique Hyperspace, in Continuum Theory: Proceedings of the Special Session in Honor of Professor Sam B. Nadler, Jr.’s 60th Birthday. Lecture Notes in Pure and Applied Mathematics Series, Vol. 230, Marcel Dekker, Inc., New York, Basel, 2002, 33–49. (eds.: Alejandro Illanes, Ira Wayne Lewis and Sergio Macías.)
D. P. Bellamy, The Cone Over the Cantor Set-continuous Maps From Both Directions, Proc. Topology Conference Emory University, Atlanta, Ga., 1970, 8–25. (ed. J. W. Rogers, Jr.)
D. E. Bennett, Aposyndetic Properties of Unicoherent Continua, Pacific J. Math., 37 (1971), 585–589.
R H Bing, Partitioning a Set, Bull. Amer. Math. Soc., 55 (1949), 1101–1110.
K. Borsuk, Theory of Retracts, Monografie Mat. Vol. 44, PWN (Polish Scientific Publishers), Warszawa, 1967.
K. Borsuk, Some Remarks on Shape Properties of Compacta, Fund. Math., 85 (1974), 185–195.
K. Borsuk and S. Ulam, On Symmetric Products of Topological Spaces, Bull. Amer. Math. Soc., 37 (1931), 875–882.
K. Borsuk and R. Molski, On a Class of Continuous Mappings, Fund. Math., 45 (1957), 84–98.
R. Bott, On the Third Symmetric Potency of \(\mathcal {S}_1\), Fund. Math., 39 (1952), 364–368.
C. E. Burgess, Chainable Continua and Indecomposability, Pacific J. Math., 9 (1959), 653–659.
J. Camargo, D. Herrera and S. Macías, Cells and n-fold Hyperspaces, to appeat in Colloquium Mathematicum.
J. Camargo and S. Macías, On Stongly Freely Decomposable and Induced Maps, Glasnik Math., 48(68), (2013), 429–442.
E. Castañeda, A Unicoherent Continuum for Which its Second Symmetric Product is not Unicoherent, Topology Proc., 23 (1998), 61–67.
E. Castañeda, Productos Simétricos, Tesis Doctoral, Facultad de Ciencias, U. N. A. M., 2003. (Spanish)
T. A. Chapman, Lecture Notes on Hilbert Cube Manifolds, C. B. M. S. Regional Conf. Series in Math., 28 (Amer. Math. Soc., Providence, RI, 1975).
J. J. Charatonik, On Fans, Dissertationes Math. (Rozprawy Mat.), 54 (1967), 1–37.
J. J. Charatonik and A. Illanes, Local Connectedness in Hyperspaces, Rocky J. Math., 36 (2006), 811–856.
J. J. Charatonik, A. Illanes, S. Macías, Induced Mappings on the Hyperspace \(\mathcal {C}_n(X)\) of a Continuum X, Houston J. Math., 28 (2002), 781–805.
J. J. Charatonik and S. Macías, Mappings of Some Hyperspaces, JP Jour. Geometry & Topology, 4(1) (2004), 53–80.
W. J. Charatonik, On the Property of Kelley in Hyperspaces, Topology, Proceedings of the International Topological Conference held in Leningrad, 1982, Lecture Notes in Math. 1060, Springer Verlag, 1984, 7–10.
D. Curtis and N. T. Nhu, Hyperspaces of Finite Subsets Which are Homeomorphic to ℵ0-dimensional Linear Metric Spaces, Topology Appl., 19 (1985), 251–260.
C. H. Dowker, Mapping Theorems for Non-compact Spaces, Amer. J. Math., 69 (1947), 200–242.
J. Dugundji, Topology, Allyn and Bacon, Inc., Boston, London, Sydney, Toronto, 1966.
C. A. Eberhart, A Note on Smooth Fans, Colloq. Math., 20 (1969), 89–90.
C. Eberhart and S. B. Nadler, Jr., Hyperspaces of Cones and Fans, Proc. Amer. Math. Soc., 77 (1979), 279–288.
J. B. Fugate, Retracting Fans onto Finite Fans, Fund. Math., 71 (1971), 113–125.
J. B. Fugate, Small Retractions of Smooth Dendroids onto Trees, Fund. Math., 71 (1971), 255–262.
J. B. Fugate, G. R. Gordh and L. Lum, Arc-smooth Continua, Trans. Amer. Math. Soc., 265 (1981), 545–561.
T. Ganea, Symmetische Potenz Topologischer Räume, Math. Nachrichten, 11 (1954), 305–316.
J. T. Goodykoontz, Jr., Hyperspaces of Arc-smooth Continua, Houston J. Math., 7 (1981), 33–41.
J. T. Goodykoontz, Jr., Arc Smoothness in Hyperspaces, Topology Appl., 15 (1983), 131–150.
J. T. Goodykoontz, Some Retractions and Deformation Retractions on 2X and \(\mathcal {C}(X)\), Topology Appl., 21 (1985), 121–133.
J. Grispolakis, S. B. Nadler, Jr. and E. D. Tymchatyn, Some Properties of Hyperspaces with Applications to Continua Theory, Can. J. Math., 31 (1979), 197–210.
J. Grispolakis and E. D. Tymchatyn, Weakly Confluent Mappings and the Covering Property of Hyperspaces, Proc. Amer. Math. Soc., 74 (1979), 177–182.
C. L. Hagopian, Indecomposable Homogeneous Plane Continua are Hereditarily Indecomposable, Trans. Amer. Math. Soc., 224 (1976), 339–350.
J. G. Hocking and G. S. Young, Topology, Dover Publications, Inc., New York, 1988.
H. Hosokawa, Induced Mappings on Hyperspaces, Tsukuba J. Math., 21 (1997), 239–250.
H. Hosokawa, Strong Size Levels of \(\mathcal {C}_n(X)\), Houston J. Math., 37 (2011), 955–965.
W. Hurewicz, Sur la Dimension des Produits Cartésiens, Annals of Math., 36 (1935), 194–197.
W. Hurewicz and H. Wallman, Dimension Theory, Princeton Univ. Press, Princeton, N. J., 1948.
A. Illanes, Multicoherence of Symmetric Products, An. Inst. Mat. Univ. Nac. Autónoma México, 25 (1985), 11–24.
A. Illanes, Monotone and Open Whitney maps, Proc. Amer. Math. Soc., 98 (1986), 516–518.
A. Illanes, Multicoherence of Whitney Levels, Topology Appl., 68 (1996), 251–265.
A. Illanes, The Hyperspace \(\mathcal {C}_2(X)\) for a Finite Graph X is Unique, Glansnik Mat., 37(57) (2002), 347–363.
A. Illanes, Comparing n-fold and m-fold Hyperspaces, Topology Appl., 133 (2003), 179–198.
A. Illanes, Finite Graphs Have Unique Hyperspaces \(\mathcal {C}_n(X)\), Topology Proc., 27 (2003), 179–188.
A. Illanes, A Model for the Hyperspace \(\mathcal {C}_2(\mathcal {S}^1)\), Q. & A. in General Topology, 22 (2004), 117–130.
A. Illanes and S. B. Nadler, Jr., Hyperspaces: Fundamentals and Recent Advances, Monographs and Textbooks in Pure and Applied Math., Vol. 216, Marcel Dekker, New York, Basel, 1999.
F. B. Jones, Certain Homogeneous Unicoherent Indecomposable Continua, Proc. Amer. Math. Soc., 2 (1951), 855–859.
J. L. Kelley, Hyperspaces of a Continuum, Trans. Amer. Math. Soc., 52 (1942), 22–36.
J. Krasinkiewicz, On the Hyperspaces of Snake-like and Circle-like Continua, Fund. Math., 84 (1974), 155–164.
J. Krasinkiewicz, On the Hyperspaces of Hereditarily Indecomposable Continua, Fund. Math., 84 (1974), 175–186.
J. Krasinkiewicz, Curves Which Are Continuous Images of Tree-like Continua Are Movable, Fun. Math., 89 (1975), 233–260.
W. Kuperberg, Uniformly Pathwise Connected Continua, in Studies in Topology, Proc. Conf. Univ. North Carolina, Charlotte, NC, 1974, Academic Press, New York, 1975, 315–324. (eds. N. M. Stavrakas and K. R. Allen.)
K. Kuratowski, Topology, Vol. II, Academic Press, New York, N. Y., 1968.
M. Levin and Y. Sternfeld, The Space of Subcontinua of a 2-dimensional Continuum is Infinitely Dimensional, Proc. Amer. Math. Soc., 125 (1997), 2771–2775.
J. C. Macías, El n-ésimo Pseudohiperespacio Suspensión de Continuos, Tesis de Doctorado, Facultad de Ciencias Físico Matemáticas, B. U. A. P., 2008. (Spanish)
S. Macías, On Symmetric Products of Continua, Topology Appl., 92 (1999), 173–182.
S. Macías, Aposyndetic Properties of Symmetric Products of Continua, Topology Proc., 22 (1997), 281–296.
S. Macías, On the Hyperspaces \(\mathcal {C}_n(X)\) of a Continuum X, Topology Appl., 109 (2001), 237–256.
S. Macías, On the Hyperspaces \(\mathcal {C}_n(X)\) of a Continuum X, II, Topology Proc., 25 (2000), 255–276.
S. Macías, On Arcwise Accessibility in Hyperspaces, Topology Proc., 26 (2001–2002), 247–254.
S. Macías, Fans Whose Hyperspaces Are Cones, Topology Proc., 27 (2003), 217–222.
S. Macías, Correction to the paper “On the Hyperspaces \(\mathcal {C}_n(X)\) of a continuum X, II”, Topology Proc., 30 (2006), 335–340.
S. Macías, On the n-fold Hyperspace Suspension of Continua, II, Glasnik Mat., 41(61) (2006), 335–343.
S. Macías, On n-fold Hyperspaces, Glasnik Mat., 44(64) (2009), 479–492.
S. Macías, Retractions and Hyperspaces, Glasnik Mat., 46(66) (2011), 473–483.
S. Macías, Deformation Retracts and Hilbert cubes in n-fold hyperspaces, Topology Proc., 40 (2012), 215–226.
S. Macías and S. B. Nadler, Jr., n-fold Hyperspaces, Cones and Products, Topology Proc., 26 (2001–2002), 255–270.
S. Macías and S. B. Nadler, Jr., Z-sets in Hyperspaces, Q. & A. in General Topology, 19 (2001), 227–241.
S. Macías and S. B. Nadler, Jr., Smoothness in n-fold Hyperspaces, Glasnik Mat., 37(57) (2002), 365–373.
S. Macías and S. B. Nadler, Jr., Fans Whose Hyperspace of Subcontinua are Cones, Topology Appl., 126 (2002), 29–36.
S. Macías and S. B. Nadler, Jr., Various Types of Local Connectedness in n-fold Hyperspaces, Topology Appl., 54 (2007), 39–53.
S. Macías and César Piceno, Strong Size Properties, Glasnik Mat., 48(68) (2013), 103–114.
S. Macías and César Piceno, More on Strong Size Properties, Glasnik Mat., 50(70) (2015), 467–488.
S. Mardešić and J. Segal, ε-mappings onto Polyhedra, Trans. Amer. Math. Soc., 109 (1963), 146–164.
M. M. Marsh, s-connected Spaces and the Fixed Point Property, Topology Proc., 8 (1983), 85–97.
V. Martínez-de-la-Vega, Dimension of n-fold Hyperspaces of Graphs, Houston J. Math., 32 (2006), 783–799.
J. M. Martínez-Montejano, Zero-dimensional Closed Set Aposyndesis and Hyperspaces, Houston J. Math., 32 (2006), 1101–1105.
J. van Mill, Infinite-Dimensional Topology, North Holland, Amsterdam, 1989.
E. E. Moise, Grille Decomposition and Convexification Theorems for Compact Locally Connected Continua, Bull. Amer. Math. Soc., 55 (1949), 1111–1121.
J. R. Munkres, Elements of Algebraic Topology, Addison-Wesley, Reading, MA, 1984.
S. B. Nadler, Jr., Arcwise Accessibility in Hyperspaces, Dissertationes Math. (Rozprawy Mat.), 138 (1976), 1–29.
S. B. Nadler, Jr., Hyperspaces of Sets, Monographs and Textbooks in Pure and Applied Math., Vol. 49, Marcel Dekker, New York, Basel, 1978. Reprinted in: Aportaciones Matemáticas de la Sociedad Matemática Mexicana, Serie Textos # 33, 2006.
S. B. Nadler, Jr., Continua Whose Hyperspace is a Product, Fund. Math., 108 (1980), 49–66.
S. B. Nadler, Jr., Continuum Theory: An Introduction, Monographs and Textbooks in Pure and Applied Math., Vol. 158, Marcel Dekker, New York, Basel, Hong Kong, 1992.
S. B. Nadler, Jr., Dimension Theory: An Introduction with Exercises, Aportaciones Matemáticas, Serie Textos # 18, Sociedad Matemática Mexicana, 2002.
L. Paredes-Rivas and P. Pellicer-Covarrubias, On Strong Size Levels, Topology Appl., 160 (2013), 1816–1828.
A. Petrus, Contractibility of Whitney Continua in \(\mathcal {C}(X)\), General Topology Appl., 9 (1978), 275–288.
J. T. Rogers, Jr., Applications of a Vietoris–Begle Theorem for Multi-valued Maps to the Cohomology of Hyperspaces, Michigan Math. J., 22 (1975), 315–319.
J. T. Rogers, Jr., Dimension and the Whitney Subcontinua of \(\mathcal {C}(X)\), Gen. Top. and its Applications, 6 (1976), 91–100.
A. H. Wallace, Algebraic Topology, Homology and Cohomology, W. A. Benjamin Inc., 1970.
L. E. Ward, Jr. A Note on Whitney Maps, Canad. Math.Bull., 23 (1980), 373–374.
L. E. Ward, Extending Whitney Maps, Pacific J. Math., 93 (1981), 465–469.
H. Whitney, On Regular Families of Curves I, Proc. Nat. Acad. Sci., 18 (1932), 275–278.
G. T. Whyburn, Analytic Topology, Amer. Math. Soc. Colloq. Publ., Vol. 28, Amer. Math. Soc., Providence, R. I., 1942.
S. Willard, General Topology, Addison-Wesley Publishing Co., 1970.
M. Wojdisławski, Rétractes Absolus et Hyperespaces des Continus, Fund. Math., 32 (1939), 184–192.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG, part of Springer Nature
About this chapter
Cite this chapter
Macías, S. (2018). n-Fold Hyperspaces. In: Topics on Continua. Springer, Cham. https://doi.org/10.1007/978-3-319-90902-8_6
Download citation
DOI: https://doi.org/10.1007/978-3-319-90902-8_6
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-90901-1
Online ISBN: 978-3-319-90902-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)