On a Topological Operator via Local Closure Function

Yıl 2023, , 227 - 236, 31.12.2023

Ayşe Nur Tunç , Sena Özen Yıldırım

https://doi.org/10.47000/tjmcs.1195540

Cited By: 1

Öz

In this research, we define and study the new topological operator called $\Gamma$-boundary operator $Bd^{\Gamma}$ by merging local closure function in ideal topological spaces. We research essential properties of this operator and we specialize $\Gamma$-boundary of some special sets, such as $\theta$-open, $\Im_{\Gamma}$-perfect and $\Im_{\Gamma}$-dense. Moreover, we examine the properties of this operator in the topology which is formed by using local closure function. Furthermore, we compare $\Gamma$-boundary operator with the boundary operator and the $*$-boundary operator. We also show that under what conditions $\Gamma$-boundary operator, boundary operator and $*$-boundary operator are coincide.

Anahtar Kelimeler

Operator $Bd^{\Gamma}$, local closure function, ideal topological space, $\Im_{\theta}$-open set

Kaynakça

• Al-Omari, A., Noiri, T., Local closure functions in ideal topological spaces, Novi Sad J. Math., 43(2)(2013), 139–149.
• Bourbaki, N., General Topology, Chapter 1-4, Springer, 1989.
• Dontchev, J., Idealization of Ganster-Reilly decomposition theorems, arXiv: math. Gn/9901017v1, 5 Jan 1999.
• Goyal, N., Noorie, N.S., $\theta$-closure and $T_{2\frac{1}{2}}$ spaces via ideals, Italian Journal of Pure and Applied Mathematics, 41(2019), 571–583.
• Kuratowski, K., Topology, Vol. I, Academic Press, New York, 1966.
• Levine, N., Generalized closed sets in topology, Rend. Circ. Mat. Palermo, 19(1970), 89–96.
• Mashhour, A.S., Abd El-Monsef, M.E., El-Deeb, S.N., On precontinuous and weak precontinuous mappings, Proc. Math. Phys. Soc. Egypt, 53(1982), 47–53.
• Natkaniec, T., On I-continuity and I-semicontinuity points, Math. Slovaca, 36(3)(1986), 297–312.
• Njamcul, A., Pavlovi´c, A., On closure compatibility of ideal topological spaces and idempotency of the local closure function, Periodica Mathematica Hungarica, 84(2)(2022), 221–234.
• Noorie, N.S., Goyal, N., On $S_{2\frac{1}{2}}$ mod I spaces and $\theta^{I}$-closed sets, International Journal of Mathematics Trends and Technology, 52(4)(2017), 226–228.
• Pavlovic, A., Local function versus local closure function in ideal topological spaces, Filomat, 30(14)(2016), 3725–3731.
• Selim, Sk., Modak, S., Islam, Md. M., Characterizations of Hayashi-Samuel spaces via boundary points, Commun. Adv. Math. Sci., 2(3)(2019), 219–226.
• Tunç, A.N., Özen Yıldırım, S., New sets obtained by local closure functions, Annals of Pure and Applied Mathematical Sciences, 1(1)(2021), 50–59.
• Tunç, A. N., Özen Yıldırım, S., A study on further properties of local closure functions, 7th International Conference on Recent Advances in Pure and Applied Mathematics (ICRAPAM 2020), (2020), 123–123.
• Vaidyanathaswamy, R., The localisation theory in set-topology, Proc. Indian Acad. Sci., Sect. A., 20(1944), 51–61.
• Velicko, N. V., H-closed topological spaces, Mat. Sb. (N.S.), 70(112)(1966), 98–112. English transl., Amer. Math. Soc. Transl., 78(2)(1968), 102–118.