Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

GENERALIZED FRÉCHET-URYSOHN SPACES

  • Hong, Woo-Chorl (Department of Mathematics Education Pusan National University)
  • Published : 2007.03.31
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we introduce some new properties of a topological space which are respectively generalizations of $Fr\'{e}chet$-Urysohn property. We show that countably AP property is a sufficient condition for a space being countable tightness, sequential, weakly first countable and symmetrizable, to be ACP, $Fr\'{e}chet-Urysohn$, first countable and semimetrizable, respectively. We also prove that countable compactness is a sufficient condition for a countably AP space to be countably $Fr\'{e}chet-Urysohn$. We then show that a countably compact space satisfying one of the properties mentioned here is sequentially compact. And we show that a countably compact and countably AP space is maximal countably compact if and only if it is $Fr\'{e}chet-Urysohn$. We finally obtain a sufficient condition for the ACP closure operator $[{\cdot}]_{ACP}$ to be a Kuratowski topological closure operator and related results.

Keywords

References

  1. A. V. Arhangel'skii, The frequency spectrum of a topological space and the product operation, Trans. Moscow Math. Soc. 2 (1981), 163-200
  2. A. V. Arhangel'skii and L. S. Pontryagin(eds.), General Topology I, Encyclopaedia of Mathematical Sciences, vol. 17, Springer-Verlage, Berlin, 1990
  3. A. Bella, On spaces with the property of weak approximation by points, Comment. Math. Univ. Carolin. 35 (1994), no. 2, 357-360
  4. A. Bella and I. V. Yaschenko, On AP and WAP spaces, Comment. Math. Univ. Carolin. 40 (1999), no. 3, 531-536
  5. J. Dugundji, Topology, Reprinting of the 1966 original. Allyn and Bacon Series in Advanced Mathematics. Allyn and Bacon, Inc., Boston, Mass.-London-Sydney, 1978
  6. S. P. Franklin, Spaces in which sequences suffice, Fund. Math. 57 (1965), 107-115 https://doi.org/10.4064/fm-57-1-107-115
  7. G. Gruenhage, Generalized metric spaces, Handbook of set-theoretic topology, 423-501, North-Holland, Amsterdam, 1984
  8. W. C. Hong, A note on weakly first countable spaces, Commun. Korean Math. Soc. 17 (2002), no. 3, 531-534 https://doi.org/10.4134/CKMS.2002.17.3.531
  9. W. C. Hong, A theorem on countably Frechet-Urysohn spaces, Kyungpook Math. J. 43 (2003), no. 3, 425-431
  10. E. A. Michael, A quintuple quotient quest, General Topology and Appl. 2 (1972), 91-138 https://doi.org/10.1016/0016-660X(72)90040-2
  11. J. Pelant, M. G. Tkachenko, V. V. Tkachuk, and R. G. Wilson, Pseudocompact Whyburn spaces need not be Frechet, Proc. Amer. Math. Soc. 131 (2003), no. 10, 3257-3265 https://doi.org/10.1090/S0002-9939-02-06840-5
  12. T. W. Rishel, A class of spaces determined by sequences with their cluster points, Portugal. Math. 31 (1972), 187-192
  13. F. Siwiec, On defining a space by a weak base, Pacific J. Math. 52 (1974), 233-245 https://doi.org/10.2140/pjm.1974.52.233
  14. L. A. Steen and J. A. Seebach, Jr., Counterexamples in topology, Second edition. Springer-Verlag, New York-Heidelberg, 1978
  15. V. V. Tkachuk and I. V. Yaschenko, Almost closed sets and topologies they determine, Comment. Math. Univ. Carolin. 42 (2001), no. 2, 395-405
  16. J. E. Vaughan, Countably compact and sequentially compact spaces, Handbook of settheoretic topology, 569-602, North-Holland, Amsterdam, 1984

Cited by

  1. A NOTE ON SPACES WHICH HAVE COUNTABLE TIGHTNESS vol.26, pp.2, 2011, https://doi.org/10.4134/CKMS.2011.26.2.297
  2. GENERALIZED PROPERTIES OF STRONGLY FRÉCHET vol.34, pp.1, 2012, https://doi.org/10.5831/HMJ.2012.34.1.85
  3. ON SPACES WHICH HAVE COUNTABLE TIGHTNESS AND RELATED SPACES vol.34, pp.2, 2012, https://doi.org/10.5831/HMJ.2012.34.2.199
  4. SOME NECESSARY AND SUFFICIENT CONDITIONS FOR A FRÉCHET-URYSOHN SPACE TO BE SEQUENTIALLY COMPACT vol.24, pp.1, 2009, https://doi.org/10.4134/CKMS.2009.24.1.145
  5. NEW CARDINAL FUNCTIONS RELATED TO ALMOST CLOSED SETS vol.35, pp.3, 2013, https://doi.org/10.5831/HMJ.2013.35.3.541
  6. ON SPACES IN WHICH THE THREE MAIN KINDS OF COMPACTNESS ARE EQUIVALENT vol.25, pp.3, 2010, https://doi.org/10.4134/CKMS.2010.25.3.477
  7. A NOTE ON SPACES DETERMINED BY CLOSURE-LIKE OPERATORS vol.32, pp.3, 2016, https://doi.org/10.7858/eamj.2016.027
  8. A NOTE ON THE FIRST LAYERS OF ℤp-EXTENSIONS vol.24, pp.1, 2009, https://doi.org/10.4134/CKMS.2009.24.1.001
  9. A GENERALIZATION OF A SEQUENTIAL SPACE AND RELATED SPACES vol.36, pp.2, 2014, https://doi.org/10.5831/HMJ.2014.36.2.425
  10. Diagonal sequence property in Banach spaces with weaker topologies vol.350, pp.2, 2009, https://doi.org/10.1016/j.jmaa.2008.08.040