A note on the new set operator \(\psi_r\). (English) Zbl 1445.54001

Summary: Recently many published works made on local function used in ideal topological spaces can be found in related literature. [A. Al-Omari and T. Noiri, Novi Sad J. Math. 43, No. 2, 139–149 (2013; Zbl 1349.54002), and S. Modak and S. Mistry, Int. J. Contemp. Math. Sci. 7, No. 1–4, 1–8 (2012; Zbl 1247.54006)] can be mentioned among such works those aim to define such functions. In general, the researchers prefer using the generalized open sets instead of topology in ideal topological spaces. Obtaining a Kuratowski closure operator with the help of local functions is an important detail in ideal topological space. However, it is not possible to obtain a Kuratowski closure operator from many of these local functions proposed by the above mentioned works. In order to address the lack of such an operator, the goal of this paper is to introduce another local function to give possibility of obtaining a Kuratowski closure operator. On the other hand, regular local functions defined for ideal topological spaces have not been found in the current literature. Regular local functions for the ideal topological spaces has been described within this work. Moreover, with the help of regular local functions Kuratowski closure operators \(cl_I^{\ast r}\) and \(\tau^{\ast r}\) topology are obtained. Many theorems in the literature have been revised according to the definition of regular local functions.


54A05 Topological spaces and generalizations (closure spaces, etc.)
54C10 Special maps on topological spaces (open, closed, perfect, etc.)
Full Text: DOI Euclid


[1] A. Vadivel and Mohanarao Navuluri, Regular semi local functions in ideal topological spaces, Journal of Advenced Research in Scientific Computing 5, (2013), 1-6. · Zbl 1338.54100
[2] A. Vadivel and K. Vairamanickam, \(rg\alpha \)-Closed Sets and \(rg\alpha \)-Open Sets in Topological Spaces, Int. Journal of Math. Analysis 3, (2009), 1803 - 1819. · Zbl 1200.54002
[3] Ahmad Al Omari and Takashi Noiri, Local closure functions in ideal topological spaces, Novi Sad J. Math. 43, (2013), 139-149. · Zbl 1349.54002
[4] D. Jankovic and T.R. Hamlett, New topologies from old via ideals, Amer. Math. Monthly 97, (1990), 295 - 310. · Zbl 0723.54005
[5] J. Dontchev, M. Ganster and D. Rose, Ideal resolvability, Topology Appl. 93(1), (1999), 1 - 16. · Zbl 0955.54001
[6] K. Kuratowski, Topology, Academic Press, NewYork 1, (1966).
[7] M. Khan and T. Noiri, Semi local functions in ideal topological spaces, Journal of Advenced Research in Pure Mathematics (2010), 36-42.
[8] Nirmala Rebecca Paul, RgI-closed sets in ideal topological spaces, International Journal of Computer Applications 69(2013), no.4.
[9] N. Levine, Semi-open sets and semi-continuity in topological spaces, Am. Math. Mon. 70(1963), 36-41. · Zbl 0113.16304
[10] O. Njastad, Remarks on topologies defined by local properties, Anh. Norske Vid. -Akad. Oslo (N.S.) 8(1966), 1-6. · Zbl 0148.16504
[11] Sukalyan Mistry and Shyamapada Modak, \(()^{*p}\) and \(\psi_p\) operator, International Mathematical Forum 7(2012), no.2, 89-96. · Zbl 1247.54005
[12] S. Modak and C. Bandyopadhyay, A note on \(\psi \) - operator, Bull. Malyas. Math. Sci. Soc. 30(2007), no. (2), 43-48. · Zbl 1133.54302
[13] T.R. Hamlett and D. Jankovic, Ideals in topological spaces and the set operator \(\psi \), Bull. U.M.I., 7, (1990), no. 4-B, 863-874. · Zbl 0741.54002
[14] R. Vaidyanathaswamy, The localization theory in set topology, Proc. Indian Acad. Sci 20, (1945), 51-61. · Zbl 0061.39308
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.