Glasnik Matematicki, Vol. 50, No. 2 (2015), 467-488.

MORE ON STRONG SIZE PROPERTIES

Sergio Macías and César Piceno

Abstract.   We continue our study of strong size maps. We show that strong size levels for the n-fold hyperspace of a continuum contain (n-1)-cells. We give two constructions of strong size maps. We introduce reversible strong size properties. We prove that each of the following properties: being a continuum chainable continuum, being a locally connected continuum, and being a continuum with the property of Kelley, is a reversible strong size property. Following Professors Goodykoontz and Nadler, we define admissible strong size maps and show that the levels of admissible strong size maps for the n-fold hyperspace of a locally connected continuum are homeomorphic to the Hilbert cube. Professor Benjamín Espinoza defined Whitney preserving maps for the hyperspace of subcontinua of a continuum. We define strong size preserving maps and show that this class of maps coincides with the class of homeomorphisms.

2010 Mathematics Subject Classification.   54B20.

Key words and phrases.   Absolute retract, acyclic continuum, admissible strong size map, continuum, continuum chainable continuum, Hilbert cube, n-fold hyperspace, n-fold symmetric product, retract, retraction, reversible strong size property, strong size level, strong size map, strong size properties.

DOI: 10.3336/gm.50.2.14

References:

1. R. H. Bing, Partitioning a set, Bull. Amer. Math. Soc. 55 (1949), 1101-1110.
   MathSciNet     CrossRef

2. K. Borsuk, Theory of retracts, Monogr. Mat. 44, Państwowe Wydawnictwo Naukowe, Warsaw, 1967.
   MathSciNet    

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

4. T. A. Chapman, Lectures on Hilbert cube manifolds, American Mathematical Society, Providence, 1976.
   MathSciNet     CrossRef

5. J. J. Charatonik, A. Illanes and S. Macías, Induced mappings on the hyperspaces C_n(X) of a continuum X, Houston J. Math. 28 (2002), 781-805.
   MathSciNet    

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

7. D. W. Curtis and R. M. Schori, Hyperspaces which characterize simple homotopy type, General Topology and Appl. 6 (1976), 153-165.
   MathSciNet     CrossRef

8. B. Espinoza Reyes, Whitney preserving functions, Topology Appl. 126 (2002), 351-358.
   MathSciNet     CrossRef

9. J. T. Goodykoontz and S. B. Nadler, Jr., Whitney levels in hyperspaces of certain Peano continua, Trans. Amer. Math. Soc. 274 (1982), 671-694.
   MathSciNet     CrossRef

10. H. Hosokawa, Induced mappings on hyperspaces, Tsukuba J. Math. 21 (1997), 239-250.
    MathSciNet    

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

12. A. Illanes, Monotone and open Whitney maps, Proc. Amer. Math. Soc. 98 (1986), 516-518.
    MathSciNet     CrossRef

13. A. Illanes, S. Macías and S. B. Nadler, Jr., Symmetric products and Q-manifolds, in Geometry and topology in dynamics, American Mathematical Society, Providence, 1999, 137-141.
    MathSciNet     CrossRef

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

15. J. W. Jaworowski, Symmetric products of ANR's, Math. Ann. 192 (1971), 173-176.
    MathSciNet     CrossRef

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

17. S. Macías and C. Piceno, Strong size properties, Glas. Mat. Ser. III 48(68) (2013), 103-114.
    MathSciNet     CrossRef

18. E. E. Moise, Grille decomposition and convexification theorems for compact metric locally connected continua, Bull. Amer. Math. Soc. 55 (1949), 1111-1121.
    MathSciNet     CrossRef

19. S. B. Nadler, Jr., Hyperspaces of sets, Marcel Dekker, New York, Basel, 1978. Reprinted in: Aportaciones Matemáticas de la Sociedad Matemática Mexicana, Serie Textos # 33, 2006.
    MathSciNet    

20. S. B. Nadler, Jr., Continuum theory: an introduction, Marcel Dekker, New York, Basel, Hong Kong, 1992.
    MathSciNet    

21. H. Toruńczyk, On CE-images of the Hilbert cube and characterization of Q-manifolds, Fund. Math. 106 (1980), 31-40.
    MathSciNet    

22. H. Whitney, On regular families of curves I, Proc. Nat. Acad. Sci. 18 (1932), 275-278.
    CrossRef

23. M. Wojdisławski, Rétractes absolus et hyperspaces des continus, Fund. Math. 32 (1939), 184-192.