Int. J. Math. And Appl., 6(2–B)(2018), 25–32 ISSN: 2347-1557 Available Online: http://ijmaa.in/ A p p l i c a t i o n s • I S S N : 2 3 4 7 - 1 5 5 7 • I n t e r n a t i o n a l J o u r n a l o f M a t h e m a t i c s A n d i t s International Journal of Mathematics And its Applications Certain Investigations on Digital Plane S. Pious Missier 1 and K. M. Arifmohammed 1, * 1 P.G. and Research Department of Mathematics, V.O.Chidambaram College (Affiliation of Manonmaiam Sundaranar University), Abishekapatti, Tirunelveli, Tamilnadu, India. Abstract: We introduce the concept of # g ˆ α-closed sets in a topological space and characterize it using * gαo-kernel and τ α -closure. Moreover, we investigate the properties of # g ˆ α-closed sets in digital plane. The family of all # g ˆ α-open sets of (Z 2 ,κ 2 ), forms an alternative topology of Z 2 . We prove that this plane (Z 2 , # g ˆ αO) is T 1/2 and T 3/4 . It is well known that the digital plane (Z 2 ,κ 2 ) is not T 1/2 , even if (Z,κ) is T 1/2 . MSC: 54A05, 54D01, 54F65, 54G05. Keywords: Preopen sets, generalized closed sets, α-open sets, * gα-closed sets, # g ˆ α-open sets, T 1/2 -spaces, digital lines and digital planes. c  JS Publication. Accepted on: 21.04.2018 1. Introduction In 1970, N. Levine [8] introduced and investigated the concept of generalized closed sets in a topological space. He studied most fundamental properties and also introduced a separation axiom T 1/2 . The digital line is typical example of a T 1/2 space [2]. After Levine’s works, many authors defined and investigated various notions to Levine’s g-closed sets and related topics [4]. In 1970, E. Khalimsky [6] introduced digital line. In 1990, K. Kopperman and R. Meyer [5] developed finite analog of the Jordan curve theorem motivated by a problem in computer graphics (cf. [5, 7]). In this paper, we introduce the concept of # g ˆ α-closed sets in a topological space and characterize it using * gαo-kernel and τ α -closure. Moreover, we investigate the properties of # g ˆ α-closed sets in digital plane. We prove that this plane (Z 2 , # g ˆ αO) is T 1/2 and T 3/4 . It is well known that the digital plane (Z 2 ,κ 2 ) is not T 1/2 , even if (Z,κ) is T 1/2 . 2. Preliminaries Throughout this paper, (X, τ ) or X denotes the topological spaces. For a subset A of X, the closure, the interior and the complement of A are denoted by cl(A), int(A) and A c respectively. We recall some basic definitions that are used in the sequel. Definition 2.1. A subset A of a topological space (X, τ ) is called α-open [10] if A ⊆ int(cl(int(A))). Moreover, A is said to be α-closed if X\A is α-open. The collection of all α-open subsets in (X, τ ) is denoted by τ α . The α-closure of a subset A is the smallest α-closed set containing A and this is denoted by τ α -cl(A) in this paper. * E-mail: arifjmc9006@gmail.com (Research Scholar)