Hereditary separability in Hausdorff continua

Authors

  • D. Daniel Lamar University
  • M. Tuncali Nipissing University

DOI:

https://doi.org/10.4995/agt.2012.1638

Keywords:

Hereditary separability, Image of compact ordered space, Locally connected continuum, Rim-separability

Abstract

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.

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Author Biographies

D. Daniel, Lamar University

Lamar University, Department of Mathematics, Beaumont, Texas 77710, USA

M. Tuncali, Nipissing University

Nipissing University, Faculty of Arts and Sciences, North Bay, Ontario P1B 8L7, Canada

References

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. 1 (2010), 43–57. http://dx.doi.org/10.4064/fm209-1-4

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), 187–193.

D. Daniel, J. Nikiel, M. Tuncali, L. B. Treybig and E. D. Tymchatyn, Concerning continua that contain no metric subcontinua, Houston J. Math. 30, no. 3 (2004), 745–750.

D. Daniel, J. Nikiel, M. Tuncali, L. B. Treybig and E. D. Tymchatyn, On perfectlynormal compacta, Questions Answers Gen. Topology 23, no. 1 (2005), 1–14.

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. http://dx.doi.org/10.4153/CMB-2005-017-4

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 on General Topology and its Applications” (Trieste, 1999). Rend. Istit. Mat. Univ. Trieste 32, suppl. 2 (2001), 89–99 (2002).

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. 24, no. 1 (1970), 45–56.

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

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Published

2012-04-15

How to Cite

[1]
D. Daniel and M. Tuncali, “Hereditary separability in Hausdorff continua”, Appl. Gen. Topol., vol. 13, no. 1, pp. 51–60, Apr. 2012.

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