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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.

MSC:

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

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