ON $P-\mathcal{I}$-OPEN SETS

Kang, Jeong-Gi;Kim, Chang-Su

  • Received : 2009.04.08
  • Accepted : 2009.07.13
  • Published : 2009.09.25


The notions of pre-local function, semi-local functions and ${\alpha}$-local functions with respect to a topology and an ideal are introduced, and several properties are investigated. Also, the concept of $P-\mathcal{I}$-open sets and $P-\mathcal{I}$-closed sets in ideal topological spaces are discussed. Relations between $\mathcal{I}$-open sets and $P-\mathcal{I}$-open sets are provided, and several properties related to $P-\mathcal{I}$-open sets, pre-local functions, semi-local functions and ${\alpha}$-local functions with respect to a topology and an ideal are investigated.


Ideal topological spaces;pre-local (semi-local, ${\alpha}$-local) functions with respect to a topology and an ideal;P-I-open (closed) sets


  1. A. Acikgoz, T. Noiri and S. Yuksel, On ${\delta}$-I-open sets and decomposition of ${\alpha}$-I-continuity. Acta. Math. Hungar. 102 (2004), no. 4, 349-357.
  2. A. Acikgoz, T. Noiri and S. Yuksel, On ${\alpha}$-I-continuous and ${\alpha}$-I-open functions. Acta Math. Hungar. 105 (2004), no. 1-2, 27-37.
  3. D. Andrljevic, Some properties of the topology of ${\alpha}$-sets, Mat. Vesnik 36 (1984), 1-10.
  4. M. Akdag. On b-I-open sets and b-I-continuous functions, Internat. J. Math. Math. Sci. 2007 Article ID 75721, 13 pages.
  5. M. Caldas, A note on some applications of ${\alpha}$-sets. Int. J. Math. Math. Sci. 2003 (2003), no. 2, 125-130.
  6. M. Caldas and J. Dontchev, On space s with hereditarily compact ${\alpha}$-topoioqies, Acta. Math. Hung. 82 (1999), 121-129.
  7. E. Hatir. A. Kesk in and T. Noiri, A note on strong ${\beta}$-I-open sets and strongly ${\beta}$-I-continous functions, Acta Math. Hungar. 108(2005), no. 1-2, 87-94.
  8. E. Hatir and T. Noiri, On decompositions of continuity via idealization, Acta Math. Hungar. 96 (2002), no. 4, 341-349.
  9. E. Hatir and T. Noiri, Weakly pre-I-open sets and decomposition of continuity, Acta Math. Hungar. 106 (2005), no. 3, 227-238.
  10. E. Hatir and T. Nolti, On semi-I-open sets and semi-I-continuous functions. Acta Math. Hungar. 107 (2005), no. 4, 345-353.
  11. E. Hatir and T. Noiri, On ${\beta}$-I-open sets and a decomposition of almost-I-continuity, Bull. Malay. Math. Sci. Soc. 29 (2006). no. 2, 119-124.
  12. S. Jafari and T. Noiri, Contra-${\alpha}$-continuous functions between topological spaces, Iranian Int. J. Sci. 2 (2001), no. 2. 153-167.
  13. S. Jafari and T. Noiri, Some remarks on weak ${\alpha}$-continuity, Far East J. Math. Sci. 6 (1998), no. 4, 619-625.
  14. S. N. Maheshwari and S. S. Thakur, On ${\alpha}$-irresolute mappings, Tamkang J. Math. 11 (1980), 209-214.
  15. S. N. Maheshwari and S. S. Thakur. On ${\alpha}$-compact spaces. Bull. Inst. Math. Acad. Sinica 13 (1985), 341-347.
  16. H. Maki, R. Devi and K. Balachandran, Generalized ${\alpha}$-closed sets in topology, Bull. Fukuoka Univ. Ed. Part III 42 (1993), 13-21.
  17. H. Maki and T. Noiri, The pasting lemma for ${\alpha}$-continuous map, Glas. Mat. 23(43) (1988), 357-363.
  18. A. S. Mashhour. I. A. Hasa nein and S. N. El-Deeb, A note on semi-continuity and precontinuity, Indian J. Pure Appl. Math. 13 (1982), no. 10, 1119-1123.
  19. A. S. Mashhour, I. A. Hasaneiu and S. N. EI-Deeb, ${\alpha}$-continuous and ${\alpha}$-open mappings, Acta Math. Hungar. 41 (1983), no. 3-4, 213-218.
  20. N. Levine, Semi-open sets and semi-continuity in topoloqicol spaces, Amer. Math. Monthly 70 (1963), 36-41.
  21. O. Njastad, On some classes of nearly open sets. Pacific J. Math. 15 (1965), 961-970.
  22. I. L. Reilly and M. K. Vamanamurthy, On ${\alpha}$-sets in topological spaces, Tamkang J. Math. 16 (1985), 7-11.
  23. J. Tong. On decomposition of continuity in topological spaces, Acta Math. Hungar. 54 (1989), no. 1-2, 51-55.
  24. D. Andrijevic, On b-open sets, Mat. Vesnik 48, (1996), 59-64.
  25. D. Jankovic, and T.R. Hamlett, New topologies from old via ideals. Amer. Math. Monthly 97(4) (1990), 295-310.
  26. M.E. Abd El-Monsef, E.F. Lashien and A.A. Nasef, On I-open sets and I-continious funtions, Kyungpook. Mathematicla. Journal 32 (1992), 21-30.
  27. Y. B. Jun, S, W. Jeong, H. J. Lee and J. W. Lee, Application of pre-open sets, Appl. General Top. 9 (2008), no. 2, 213-228.