A NOTE ON SPACES DETERMINED BY CLOSURE-LIKE OPERATORS

DOI QR코드

DOI QR Code

Hong, Woo Chorl;Kwon, Seonhee

  • 투고 : 2016.02.25
  • 심사 : 2016.04.21
  • 발행 : 2016.05.31

초록

In this paper, we study some classes of spaces determined by closure-like operators $[{\cdot}]_s$, $[{\cdot}]_c$ and $[{\cdot}]_k$ etc. which are wider than the class of $Fr{\acute{e}}chet-Urysohn$ spaces or the class of sequential spaces and related spaces. We first introduce a WADS space which is a generalization of a sequential space. We show that X is a WADS and k-space iff X is sequential and every WADS space is C-closed and obtained that every WADS and countably compact space is sequential as a corollary. We also show that every WAP and countably compact space is countably sequential and obtain that every WACP and countably compact space is sequential as a corollary. And we show that every WAP and weakly k-space is countably sequential and obtain that X is a WACP and weakly k-space iff X is sequential as a corollary.

키워드

sequential;$Fr{\acute{e}}chet-Urysohn$;countable tightness;k-space;AP;WAP;WACP;WADS;countably sequential

참고문헌

  1. A. V. Arhangel'skii, A characterization of very k-spaces, Czechoslovak Math. J. 18(93)(1968), 392-395.
  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. V. Arhangel'skii, Topological function spaces, Mathematics and its application, Vol.78, Kluwer Academic Publishers, 1992.
  4. A. V. Arhangel'skii and D. N. Stavrova, On a common generalization of k-spaces and spaces with countable tightness, Top. and its Appl. 51(1993), 261-268. https://doi.org/10.1016/0166-8641(93)90081-N
  5. A. Bella, On spaces with the property of weak approximation by points, Comment. Math. Univ. Carolinae 35(2)(1994), 357-360.
  6. A. Bella and I. V. Yaschenko, On AP andWAP spaces, Comment. Math. Univ. Carolinae 40(3)(1999), 531-536.
  7. A. Dow, M. G. Tkachenko, V. V. Tkachuk and R. G .Wilson, Topologies generated by discrete subspaces, Glansnik Math. 37(57)(2002), 187-210.
  8. S. P. Franklin, Spaces in which sequences suffice, Fund. Math. 57(1965), 107-115. https://doi.org/10.4064/fm-57-1-107-115
  9. W. C. Hong, Generalized Frechet-Urysohn spaces, J. Korean Math. Soc. 44(2)(2007), 261-273. https://doi.org/10.4134/JKMS.2007.44.2.261
  10. W. C. Hong, On spaces in which compact-like sets are closed, and related spaces, Commun. Korean Math. Soc. 22(2)(2007), 297-303. https://doi.org/10.4134/CKMS.2007.22.2.297
  11. W. C. Hong, On spaces which have countable tightness and related spaces, Honam Math. J. 34(2)(2012), 199-208. https://doi.org/10.5831/HMJ.2012.34.2.199
  12. W. C. Hong and S. Kwon, A generalization of a sequential space and related spaces, Honam Math. J. 36(2)(2014), 425-434. https://doi.org/10.5831/HMJ.2014.36.2.425
  13. M. Ismail and P. Nyikos, On spaces in which countably compact sets are closed, and hereditary properties, Top. and its Appl. 11(1980), 281-292. https://doi.org/10.1016/0166-8641(80)90027-9
  14. S. Lin and C. Zheng, The k-quotient images of metric spaces, Commun. Korean Math. Soc. 27(2012), 377-384. https://doi.org/10.4134/CKMS.2012.27.2.377
  15. M. A. Moon, M. H. Cho and J. Kim, On AP spaces in concern with compact-like sets and submaximality, Comment. Math. Univ. Carolinae 52(2)(2011), 293-302.
  16. 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
  17. Y. Tanaka, Necessary and sufficient conditions for products of k-spaces, Top. Proceedings 14(1989), 281-313.
  18. V. V. Tkachuk and I. V. Yaschenko, Almost closed sets and topologies they determine, Comment. Math. Univ. Carolinae 42(2)(2001), 395-405.
  19. A. Wilansky, Topology for analysis, Ginn and Company 1970.

과제정보

연구 과제 주관 기관 : Pusan National University