Hereditary separability in Hausdorff continua

• Daniel, D. ^[1] ; Tuncali, M. ^[2]
  1. [1] Lamar University

     Lamar University

     Estados Unidos

     
  2. [2] Nipissing University

     Nipissing University

     Canadá

     
• Localización: Applied general topology, ISSN-e 1989-4147, ISSN 1576-9402, Vol. 13, Nº. 1, 2012, págs. 51-60
• Idioma: inglés
• DOI: 10.4995/agt.2012.1638
• Enlaces
  • Texto completo
• Resumen

  • We consider those Hausdorff continua S such that each separable subspace of S is hereditarily separable. Due to results of Ostaszewski and Rudin, respectively, all monotonically normal spaces and therefore all continuous Hausdorff images of ordered compacta also have this property. Our study focuses on the structure of such spaces that also possess one of various rim properties, with emphasis given to rim-separability. In so doing we obtain analogues of results of M. Tuncali and I. Loncar, respectively.

• Referencias bibliográficas
  • A. V. Arkhangelskii, On cardinal invariants, Proceed. Third Prague Top. Symp. 1971, Prague, 1972, 37–46.
  • T. Banakh, V.V. Fedorchuk, J. Nikiel and M. Tuncali, The Suslinian number and other cardinal invariants of continua, Fund. Math. 209, no....
  • D. E. Cameron, On subspaces of separable spaces, Math. Mag. 48, no. 5 (1975), 288.
  • D. Daniel, On metrizability of images of ordered compacta, Houston J. Math. 32, no. 4 (2006), 1047–1059.
  • D. Daniel, J. Nikiel, L. B. Treybig, H. M. Tuncali and E. D. Tymchatyn, On rimproperties of continua, Questions Answers Gen. Topology 19 (2001),...
  • D. Daniel, J. Nikiel, M. Tuncali, L. B. Treybig and E. D. Tymchatyn, Concerning continua that contain no metric subcontinua, Houston J. Math....
  • D. Daniel, J. Nikiel, M. Tuncali, L. B. Treybig and E. D. Tymchatyn, On perfectlynormal compacta, Questions Answers Gen. Topology 23, no....
  • D. Daniel, J. Nikiel, L. B. Treybig, H. M. Tuncali and E. D. Tymchatyn, On Suslinian continua, Canad. Math. Bull. 48, no. 2 (2005), 195–202....
  • D. Daniel and L. B. Treybig, A decomposition theorem for locally connected Suslinian continua, Topology Proc. 23 (1998), 93–105.
  • R. W. Heath, D. J. Lutzer and P. L. Zenor, Monotonically normal spaces, Trans. Amer. Math. Soc. 178 (1973), 481–493. http://dx.doi.org/10.1090/S0002-9947-1973-0372826-2
  • E. Induráin, The concept of separable connectedness: applications to general utility theory, Proceedings of the “II Italian-Spanish Congress...
  • I. Loncar, The Property of Kelley in nonmetric continua, Math. Comm. 5 (2000), 41–50.
  • I. Loncar, Non-metric rim-metrizable continua and unique hyperspace, Publ. Inst. Math. 73(87) (2003), 97–113. http://dx.doi.org/10.2298/PIM0373097L
  • V. I. Malykhin, A non-hereditarily separable space with separable closed subspaces, Questions Answers Gen. Topology 12, no. 2 (1994), 209–214.
  • S. Mardesic, Locally connected, ordered and chainable continua, Rad. JAZU 33, no. 4 (1960), 147–166.
  • S. B. Nadler, Hyperspaces of sets, Marcel Dekker, Inc., New York, 1978.
  • J. Nikiel, The Hahn-Mazurkiewicz Theorem for hereditarily locally connected continua, Topology Appl. 32 (1989), 307–323. http://dx.doi.org/10.1016/0166-8641(89)90037-0
  • J. Nikiel, S. Purisch and L. B. Treybig, Separable zero-dimensional spaces which are continuous images of ordered compacta, Houston J. Math....
  • J. Nikiel, M. Tuncali and E. D. Tymchatyn, Continuous Images of Arcs and Inverse Limit Methods, Mem. Amer. Math. Soc. 104, no. 498 (1993).
  • A. J. Ostaszewski, Monotone Normality and Gδ diagonals in the class of inductively generated spaces, Topology 23 (1978), 905–930.
  • Z. M. Rakowski, Monotone decompositions of hereditarily smooth continua, Fund. Math. 114 (1981), 119–125.
  • M. Rudin, Nikiel’s conjecture, Topology Appl. 116, no. 3 (2001), 305–331. http://dx.doi.org/10.1016/S0166-8641(01)00218-8
  • B. E. Shapirovskii, On the density of topological spaces, Dokl. Akad. Nauk SSSR 206 (1972), 559–562.
  • J. Simone, Metric components of continuous images of ordered compacta, Pacific J. Math. 69 (1977), 269–274. http://dx.doi.org/10.2140/pjm.1977.69.269
  • L. B. Treybig, Concerning continuous images of compact ordered spaces, Proc. Amer. Math. Soc. 15, no. 6 (1964), 866–871. http://dx.doi.org/10.1090/S0002-9939-1964-0167953-9
  • L. B. Treybig, Concerning continua which are continuous images of compact ordered spaces, Duke J. Math. 32 (1965), 417–422. http://dx.doi.org/10.1215/S0012-7094-65-03241-2
  • L. B. Treybig, Local connectivity and metrizability in connected images of ordered compacta, Glasnik Mat. 14, no. 34 (1979), 375–380.
  • M. Tuncali, Concerning continuous images of rim-metrizable continua, Proc. Amer. Math. Soc. 113, no. 2 (1991), 461–470. http://dx.doi.org/10.1090/S0002-9939-1991-1069694-9