DOI QR코드

DOI QR Code

ON $\mathcal{I}$-SCATTERED SPACES

  • Li, Zhaowen (School of Science Guangxi University for Nationalities) ;
  • Lu, Shizhan (School of Science Guangxi University for Nationalities)
  • Received : 2013.01.22
  • Published : 2014.05.31

Abstract

In this paper, $\mathcal{I}$-scattered spaces are introduced, and their characterizations and properties are given. We prove that (X, ${\tau}$) is scattered if and only if (X, ${\tau}$, $\mathcal{I}$) is $\mathcal{I}$-scattered for any ideal $\mathcal{I}$ on X.

Keywords

References

  1. G. Artico, U. Marconi, R. Moresco, and J. Pelant, Selectors and scattered spaces, Topology Appl. 111 (2001), no. 1-2, 105-134. https://doi.org/10.1016/S0166-8641(99)00186-8
  2. H. R. Bennett and J. Chaber, Scattered spaces and the class MOBI, Proc. Amer. Math. Soc. 106 (1989), no. 1, 215-221.
  3. G. Bezhanishvili, R. Mines, and P. J. Morandi, Scattered, Hausdorff-reducible, and hereditarily irresolvable spaces, Topology Appl. 132 (2003), no. 3, 291-306. https://doi.org/10.1016/S0166-8641(03)00039-7
  4. Z. Cai, D. Zheng, Z. Li, and H. Chen, I-separability on ideal topological spaces, J. Adv. Res. Pure Math. 3 (2011), no. 4, 85-91. https://doi.org/10.5373/jaram.267.100710
  5. J. Dontchev, On Hausdorff spaces via topological ideals and I-irresolute functions, Papers on general topology and applications (Slippery Rock, PA, 1993), 28-37, Ann. New York Acad. Sci., 767, New York Acad. Sci., New York, 1995.
  6. J. Dontchev, M. Ganster, and T. Noiri, Unified operation approach of generalized closed sets via topological ideals, Math. Japon. 49 (1999), no. 3, 395-401.
  7. J. Dontchev, M. Ganster, and D. Rose, Ideal resolvability, Topology Appl. 93 (1999), no. 1, 1-16. https://doi.org/10.1016/S0166-8641(97)00257-5
  8. E. Ekici, On I-Alexandroff and $I_g$-Alexandroff ideal topological spaces, Filomat 25 (2011), no. 4, 99-108. https://doi.org/10.2298/FIL1104099E
  9. S. Fujii, K. Miyazaki, and T. Nogura, Vietoris continuous selections on scattered spaces, J. Math. Soc. Japan. 54 (2002), no. 2, 273-281. https://doi.org/10.2969/jmsj/05420273
  10. E. Hatir, A. Keskin, and T. Noiri, A note on strong ${\beta}$-I-sets and strongly ${\beta}$-I-continuous functions, Acta Math. Hungar. 108 (2005), no. 1-2, 87-94. https://doi.org/10.1007/s10474-005-0210-2
  11. E. Hayashi, Topologies defined by local properties, Math. Ann. 156 (1964), 205-215. https://doi.org/10.1007/BF01363287
  12. M. Henriksen, R. Raphael, and R. G. Woods, SP-scattered spaces; a new generalization of scattered spaces, Comment. Math. Univ. Carolin. 48 (2007), no. 3, 487-505.
  13. D. S. Jankovic and T. R. Hamlett, New topologies from old via ideals, Amer. Math. Monthly 97 (1990), no. 4, 295-310. https://doi.org/10.2307/2324512
  14. V. Kannan and M. Rajagopalan, Scattered spaces, Proc. Amer. Math. Soc. 43 (1974), 402-408. https://doi.org/10.1090/S0002-9939-1974-0334150-X
  15. V. Kannan and M. Rajagopalan, Scattered spaces II, Illinois J. Math. 21 (1977), no. 4, 735-751.
  16. A. Keskin, T. Noiri, and S. Yuksel, Idealization of a decomposition theorem, Acta Math. Hungar. 102 (2004), no. 4, 269-277. https://doi.org/10.1023/B:AMHU.0000024677.08811.6a
  17. K. Kuratowski, Topology, Academic Press, New York, 1966.
  18. Z. Li and F. Lin, On I-Baire spaces, Filomat 27 (2013), no. 2, 301-310. https://doi.org/10.2298/FIL1302301L
  19. M. N. Mukherjee, B. Roy, and R. Sen, On extensions of topological spaces in terms of ideals, Topology Appl. 154 (2007), no. 18, 3167-3172. https://doi.org/10.1016/j.topol.2007.08.014
  20. R. L. Newcomb, Topologies which are compact modulo an ideal, Ph.D. thesis, University of Cal. at Santa Barbara, 1967.
  21. V. Renuka Devi, D. Sivaraj, and T. Tamizh Chelvam, Codense and completely codense ideals, Acta Math. Hungar. 108 (2005), no. 3, 197-205. https://doi.org/10.1007/s10474-005-0220-0
  22. R. Vaidyanathaswamy, The localisation theory in set topology, Proc. Indian Acad. Sci. 20 (1944), 51-61.