Glasnik Matematicki, Vol. 48, No. 1 (2013), 103-114.

STRONG SIZE PROPERTIES

Sergio Macías and César Piceno

Instituto de Matemáticas, Universidad Nacional Autónoma de México, Circuito Exterior, Ciudad Universitaria, México D. F., C. P. 04510, México
e-mail: sergiom@matem.unam.mx
e-mail: cesarpicman@hotmail.com


Abstract.   We prove that countable aposyndesis, finite-aposyndesis, continuum chainability, acyclicity (for n≥ 3), and acyclicity for locally connected continua are strong size properties. As a consequence of our results we obtain that arcwise connectedness is a strong size property which is originally proved by Hosokawa.

2010 Mathematics Subject Classification.   54B20.

Key words and phrases.   Absolute retract, acyclic continuum, continuum, continuum chainable continuum, countable aposyndesis, deformation retract, finite aposyndesis, n-fold hyperspace, retract, retraction, strong size level, strong size map, strong size properties.


Full text (PDF) (access from subscribing institutions only)

DOI: 10.3336/gm.48.1.10


References:

  1. K. Borsuk and S. Ulam, On symmetric products of topological spaces, Bull. Amer. Math. Soc. 37 (1931), 875-882.
    MathSciNet     CrossRef

  2. J. J. Charatonik and S. Macías, Mappings of some hyperspaces, JP J. Geom. Topol. 4 (2004), 53-80.
    MathSciNet    

  3. C. H. Dowker, Mapping theorems for non-compact spaces, Amer. J. Math. 69 (1947), 200-242.
    MathSciNet     CrossRef

  4. T. Ganea, Symmetrische Potenzen topologischer Räume, Math. Nachr. 11 (1954), 305-316.
    MathSciNet     CrossRef

  5. H. Hosokawa, Strong size levels of Cn(X), Houston J. Math. 37 (2011), 955-965.
    MathSciNet    

  6. A. Illanes, Multicoherence of Whitney levels, Topology Appl. 68 (1996), 251-265.
    MathSciNet     CrossRef

  7. A. Illanes, Countable closed set aposyndesis and hyperspaces, Houston J. Math. 23 (1997), 57-64.
    MathSciNet    

  8. A. Illanes and S. B. Nadler, Jr., Hyperspaces. Fundamentals and recent advances, Marcel Dekker, New York, 1999.
    MathSciNet    

  9. S. Macías, On symmetric products of continua, Topology Appl. 92 (1999), 173-182.
    MathSciNet     CrossRef

  10. S. Macías, Aposyndetic properties of symmetric products of continua, Topology Proc. 22 (1997), 281-296.
    MathSciNet    

  11. S. Macías, Topics on continua, Chapman && Hall/CRC, Boca Raton, 2005.
    MathSciNet     CrossRef

  12. S. Macías, Deformation retracts and Hilbert cubes in n-fold hyperspaces, Topology Proc. 40 (2012), 215-226.
    MathSciNet    

  13. S. Mardešić and J. Segal, ε-mappings onto polyhedra, Trans. Amer. Math. Soc. 109 (1963), 146-164.
    MathSciNet     CrossRef

  14. J. R. Munkres, Elements of algebraic topology, Addison-Wesley, Menlo Park, 1984.
    MathSciNet    

  15. S. B. Nadler, Jr., Hyperspaces of sets, Sociedad Matemática Mexicana, México, 2006.
    MathSciNet    

  16. A. Petrus, Contractibility of Whitney continua in C(X), General Topology Appl. 9 (1978), 275-288.
    MathSciNet     CrossRef

  17. 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.
    MathSciNet     CrossRef

  18. A. H. Wallace, Algebraic topology, homology and cohomology, W. A. Benjamin, New York, 1970.
    MathSciNet    

  19. L. E. Ward, Jr., Extending Whitney maps, Pacific J. Math. 93 (1981), 465-469.
    MathSciNet     CrossRef

  20. G. T. Whyburn, Analytic Topology, AMS, New York, 1942.
    MathSciNet    

Glasnik Matematicki Home Page