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Hyperspace retractions for curves

Seria
Rozprawy Matematyczne tom/nr w serii: 370 wydano: 1997
Zawartość
Warianty tytułu
Abstrakty
EN
Abstract
We study retractions from the hyperspace of all nonempty closed subsets of a given continuum onto the continuum (which is naturally embedded in the hyperspace). Some necessary and some sufficient conditions for the existence of such a retraction are found if the continuum is a curve. It is shown that the existence of such a retraction for a curve implies that the curve is a uniformly arcwise connected dendroid, and that a universal smooth dendroid admits such a retraction. The existence of this retraction for a given dendroid implies that the dendroid admits a mean. An example of a (nonplanable) smooth dendroid that admits no mean is constructed. Some related results are obtained and open problems are stated. The results answer several questions asked in the literature.
EN
CONTENTS
1. Introduction...................................5
2. Preliminaries.................................7
3. Hyperspace retractions.................9
4. Applications to selections............15
5. Applications to means.................18
References.....................................32
Miejsce publikacji
Warszawa
Copyright
Seria
Rozprawy Matematyczne tom/nr w serii: 370
Liczba stron
34
Liczba rozdzia³ów
Opis fizyczny
Dissertationes Mathematicae, Tom CCCLXX
Daty
wydano
1997
otrzymano
1996-04-30
poprawiono
1997-04-30
Twórcy
  • Mathematical Institute, University of Wrocław, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland , jjc@math.uni.wroc.pl
  • Instituto de Matemáticas, UNAM, Circuito Exterior, Ciudad Universitaria, 04510 México, D.F., México
  • Mathematical Institute, University of Wrocław, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland, wjcharat@math.uni.wroc.pl
  • Departamento de Matemáticas, Facultad de Ciencias, UNAM, Circuito Exterior, Ciudad Universitaria, 04510 México, D.F., México
  • Mathematical Institute, University of Wrocław, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland, komil@math.uni.wroc.pl
Bibliografia
  • [1] G. Aumann, Aufbau von Mittelwerten mehrerer Argumente I, Math. Ann. 109 (1934), 235-253.
  • [2] G. Aumann, Aufbau von Mittelwerten mehrerer Argumente II (Analytische Mittelwerte), Math. Ann. 111 (1935), 713-730.
  • [3] G. Aumann, Über Räume mit Mittelbildungen, Math. Ann. 119 (1943), 210-215.
  • [4] P. Bacon, An acyclic continuum that admits no mean, Fund. Math. 67 (1970), 11-13.
  • [5] M. Bell and S. Watson, Not all dendroids have a mean, Houston J. Math. 22 (1996), 39-50.
  • [6] D. P. Bellamy, The cone over the Cantor set-continuous maps from both directions, in: Proceedings of the Topology Conference, Emory University, Atlanta, Ga., 1970, 8-25.
  • [7] K. Borsuk, Quelques théorèmes sur les ensembles unicohérents, Fund. Math. 17 (1931), 171-209.
  • [8] K. Borsuk, Theory of Retracts, PWN, Warszawa, 1967.
  • [9] J. H. Case and R. E. Chamberlin, Characterizations of tree-like continua, Pacific J. Math. 10 (1960), 73-84.
  • [10] J. J. Charatonik, Two invariants under continuity and the incomparability of fans, Fund. Math. 53 (1964), 187-204.
  • [11] J. J. Charatonik, Confluent mappings and unicoherence of continua, Fund. Math. 56 (1964), 213-220.
  • [12] J. J. Charatonik, Problems and remarks on contractibility of curves, in: General Topology and its Relations to Modern Analysis and Algebra IV, Proc. Fourth Prague Topological Symposium, 1976; Part B, Contributed Papers, Society of Czechoslovak Mathematicians and Physicists, Prague, 1977, 72-76.
  • [13] J. J. Charatonik, Contractibility and continuous selections, Fund. Math. 108 (1980), 109-118.
  • [14] J. J. Charatonik, Mapping properties of hereditary classes of acyclic curves, Period. Math. Hungar. 18 (2) (1987), 143-149.
  • [15] J. J. Charatonik, Some problems on selections and contractibility, Rend. Circ. Mat. Palermo (2) Suppl. 18 (1988), 27-30.
  • [16] J. J. Charatonik, On continuous selections for the hyperspace of subcontinua, in: Topology (Pécs 1989), Colloq. Math. Soc. János Bolyai 55, North-Holland, Amsterdam, 1993, 91-100.
  • [17] J. J. Charatonik, Contractibility of curves, Matematiche (Catania) 46 (1991), 559-592.
  • [18] J. J. Charatonik and W. J. Charatonik, Monotoneity relative to a point and inverse limits of continua, Glasnik Mat. 20 (40) (1985), 139-151.
  • [19] J. J. Charatonik, W. J. Charatonik and S. Miklos, Confluent mappings of fans, Dissertationes Math. (Rozprawy Mat.) 301 (1990).
  • [20] J. J. Charatonik and C. Eberhart, On smooth dendroids, Fund. Math. 67 (1970), 297-322.
  • [21] J. J. Charatonik and C. Eberhart, On contractible dendroids, Colloq. Math. 25 (1972), 89-98.
  • [22] J. J. Charatonik and Z. Grabowski, Homotopically fixed arcs and contractibility of dendroids, Fund. Math. 100 (1978), 229-239.
  • [23] W. J. Charatonik, Inverse limits of smooth continua, Comment. Math. Univ. Carolin. 23 (1982), 183-191.
  • [24] D. W. Curtis, A hyperspace retraction theorem for a class of half-line compactifications, Topology Proc. 11 (1986), 29-64.
  • [25] S. T. Czuba, On dendroids with Kelley's property, Proc. Amer. Math. Soc. 102 (1988), 728-730.
  • [26] R. B. Fraser, A new characterization of Peano continua, Comment. Math. Prace Mat. 16 (1972), 247-248.
  • [27] J. B. Fugate, G. R. Gordh, Jr. and L. Lum, Arc-smooth continua, Trans. Amer. Math. Soc. 265 (1981), 545-561.
  • [28] G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. Mislowe and D. S. Scott, A Compendium of Continuous Lattices, Springer, 1980.
  • [29] J. T. Goodykoontz, Jr., Connectedness im kleinen and local connectedness in $2^X$ and C(X), Pacific J. Math. 53 (1974), 387-397.
  • [30] J. T. Goodykoontz, More on connectedness im kleinen and local connectedness in C(X), Proc. Amer. Math. Soc. 65 (1977), 357-364.
  • [31] J. T. Goodykoontz, Some functions on hyperspaces of hereditarily unicoherent continua, Fund. Math. 95 (1977), 1-10.
  • [32] J. T. Goodykoontz, C(X) is not necessarily a retract of $2^X$, Proc. Amer. Math. Soc. 67 (1977), 177-178.
  • [33] J. T. Goodykoontz, A nonlocally connected continuum X such that C(X) is a retract of $2^X$, Proc. Amer. Math. Soc. 91 (1984), 319-322.
  • [34] J. T. Goodykoontz, Some retractions and deformation retractions on $2^X$ and C(X), Topology Appl. 21 (1985), 121-133.
  • [35] J. T. Goodykoontz, Jr. and S. B. Nadler, Jr., Whitney levels in hyperspaces of certain Peano continua, Trans. Amer. Math. Soc. 274 (1982), 671-694.
  • [36] B. G. Graham, On contractible fans, Fund. Math. 111 (1981), 77-93.
  • [37] S. T. Hu, Theory of Retracts, Wayne State Univ. Press, 1965.
  • [38] A. Illanes, A continuum X which is a retract of C(X) but not of $2^X$, Proc. Amer. Math. Soc. 100 (1987), 199-200.
  • [39] A. Illanes, Two examples concerning hyperspace retraction, Topology Appl. 29 (1988), 67-72.
  • [40] K. Kawamura and E. D. Tymchatyn, Continua which admit no mean, Colloq. Math. 71 (1996), 97-105.
  • [41] J. L. Kelley, Hyperspaces of a continuum, Trans. Amer. Math. Soc. 52 (1942), 22-36.
  • [42] R. J. Koch and I. S. Krule, Weak cutpoint ordering on hereditarily unicoherent continua, Proc. Amer. Math. Soc. 11 (1960), 679-681.
  • [43] A. N. Kolmogoroff, Sur la notion de la moyenne, Atti Accad. Naz. Lincei Rend. (6) 12 (1930), 388-391.
  • [44] J. Krasinkiewicz, Curves which are continuous images of tree-like continua are movable, Fund. Math. 89 (1975), 233-260.
  • [45] W. Kuperberg, Uniformly pathwise connected continua, in: Studies in Topology, Proc. Conf. Univ. North Carolina, Charlotte, N.C., 1974, Academic Press, New York, 1975, 315-324.
  • [46] K. Kuratowski, Topology, Vol. 1, Academic Press and PWN, 1966.
  • [47] K. Kuratowski, Topology, Vol. 2, Academic Press and PWN, 1968.
  • [48] K. Kuratowski, S. B. Nadler, Jr. and G. S. Young, Continuous selections on locally compact separable metric spaces, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 18 (1970), 5-11.
  • [49] T. J. Lee, Every contractible fan has the bend intersection property, Bull. Polish Acad. Sci. Math. 36 (1988), 413-417.
  • [50] T. J. Lee, Bend intersection property and dendroids of type N, Period. Math. Hungar. 23 (2) (1991), 121-127.
  • [51] A. Lelek, On weakly chainable continua, Fund. Math. 51 (1962), 271-282.
  • [52] L. Lum, Weakly smooth dendroids, Fund. Math. 83 (1974), 111-120.
  • [53] T. Maćkowiak, Confluent mappings and smoothness of dendroids, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 21 (1973), 719-725.
  • [54] T. Maćkowiak, Continuous selections for C(X), Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 26 (1978), 547-551.
  • [55] T. Maćkowiak, Contractible and nonselectible dendroids, Bull. Polish Acad. Sci. Math. 33 (1985), 321-324.
  • [56] H. C. Miller, On unicoherent continua, Trans. Amer. Math. Soc. 69 (1950), 179-194.
  • [57] L. Mohler, A characterization of smoothness in dendroids, Fund. Math. 67 (1970), 369-376.
  • [58] L. Mohler and J. Nikiel, A universal smooth dendroid answering a question of J. Krasinkiewicz, Houston J. Math. 14 (1988), 535-541.
  • [59] S. B. Nadler, Jr., Inverse limits and multicoherence, Bull. Amer. Math. Soc. 76 (1970), 411-414.
  • [60] S. B. Nadler, Jr., Some problems concerning hyperspaces, in: Lecture Notes in Math. 375, Springer, 1974, 190-194.
  • [61] S. B. Nadler, Jr., A characterization of locally connected continua by hyperspace retractions, Proc. Amer. Math. Soc. 67 (1977), 167-176.
  • [62] S. B. Nadler, Jr., Hyperspaces of Sets, M. Dekker, 1978.
  • [63] S. B. Nadler, Jr. and L. E. Ward, Jr., Concerning continuous selections, Proc. Amer. Math. Soc. 25 (1970), 369-374.
  • [64] L. G. Oversteegen, Non-contractibility of continua, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 26 (1978), 837-840.
  • [65] L. G. Oversteegen, Internal characterization of contractibility of fans, Bull. Acad. Polon. Sci. Sér. Sci. Math. 27 (1979), 385-389.
  • [66] K. Sigmon, A note on means in Peano continua, Aequationes Math. 1 (1968), 85-86.
  • [67] L. E. Ward, Jr., Rigid selections and smooth dendroids, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 19 (1971), 1041-1044.
  • [68] M. Wojdysławski, Sur la contractilité des hyperespaces des continus localement connexes, Fund. Math. 30 (1938), 247-252.
  • [69] M. Wojdysławski, Rétractes absolus et hyperespaces des continus, Fund. Math. 32 (1939), 184-192.
Języki publikacji
EN
Uwagi
1991 Mathematics Subject Classification: 54B20, 54C15, 54F15, 54F50.
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0012-3862
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DML-PL
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