On Connectedness via a G-method and a Hereditary Class

Yıl 2020, Cilt: 8 Sayı: 2, 370 - 375, 27.10.2020

Faruk Gürcan Ayşegül Çaksu Güler

Öz

In 2003, Connor and Grosse-Erdmann [1] introduced the definition of $G$-method by using G-linear functions instead of limit, based on various types of convergence on real numbers. Later on, some mathematicians examined this concept in topological groups. Then new concepts, which were important in topology such as $G$-sequential compactness and $G$-sequential connected, were defined and some properties of those concepts are investigated. S. Lin and L. Liu defined $G$-method notion by taking any set instead of topological group in 2016. In this paper, we give definition of $cl_{G^{*}}$-closure which is more general than $G$-closure of a set with the help of hereditarily class. Then we define the notion of $\tau_{G^{*}}$-topology and give the concepts of $G^{*}$-connected and $G^{*}$-component. Besides, we examine the relationship between these concepts and previously given concepts.

Anahtar Kelimeler

hereditary class, G-method, G*-closed, G*-connected, G*-component

Destekleyen Kurum

Ege University Scientific Research Projects Coordination Unit.

Proje Numarası

FYL-2020-21056

Kaynakça

• [1] J. Connor and K. Grose-Erdmann, Sequential definitions of continuty for real functions, Rocky Mt. 33(1) (2003) 93-126.
• [2] A. Csaszar, Generalized topology, generalized continuity, Acta Math. Hung. 96 (2002) 351-357.
• [3] A. Csaszar, Modification of generalized topologies via hereditarly classes, Acta Math. Hung. 115(1-2) (2007) 29-36.
• [4] H. Çakallı, Sequential definitions of compactness, App.Math.Lett. 21(6) (2008) 594-598.
• [5] H. Çakallı, Sequential definitions of connectedness, Appl. Math. Lett. 25 (2012) 461-465.
• [6] H. Fast, Sur la convergence statistique, Colloq. Math. 2 (1951), 241-244.
• [7] P. Kostyrko, T. Salat, W. Wilczynski, I-convergence, Real Anal. Exchange, 26:2 (2000-2001), 669–686.
• [8] K. Kuratowski, Topologie I, PWN, Warszawa, 1961.
• [9] S. Lin, L. Liu , G-methods, G-sequential space and G- continuity in topological spaces, Topology App. 212 (2016), 29-48.
• [10] L. Liu, G-neighborhoods in topological spaces, J. Minnan Normal Univ. Nat. Sci., 29(3) (2016) 1-6.
• [11] L. Liu, G-derived sets and G-boundary, J. Yangzhou Univ. Nat. Sci., 20(1) (2017) 18-22.
• [12] L. Liu, G-Kernel-Open Sets, G-Kernel-Neighborhoods and G-Kernel-Derived Sets, Journal of Mathematical Research with Appl. 38(3), (2018) 276-286.
• [13] G. Di. Maio, And Lj D. R. Kocinac., Statistical convergence in topology, Topology Appl., 156(1) (2008), 28-45.
• [14] O. Mucuk and T. Sahan, On G-sequental continuity,28(6) (2014) Filomat, 1181-1189.
• [15] B. Schweizer and A. Sklar, Statistical Metric Spaces , Pacific J. Maths., 10 (1960), 313-334.
• [16] H. Steinhaus, Sur la convergence ordinaire et la convergence asymptotique, Colloq. Math. 2 (1951), 73-34.
• [17] Z. Tang and F. Lin, Statistical versions of sequential and Fr˘echet-Urysohn spaces., Adv. Math. (China) 44(6) (2015), 945-954.
• [18] Y. Wu and F. Lin, The G-connected property and G-topological groups, 33(14) (2019) Filomat, 4441-4450.
• [19] A. Zygmund, Trigonometric Series, Cambridge Univ. Press, New York.