NTMSCI 6, No. 2, 119-129 (2018) 119 New Trends in Mathematical Sciences http://dx.doi.org/10.20852/ntmsci.2018.277 Some topological properties of double Ces` aro-Orlicz sequence spaces Cenap Duyar 1 and Oguz Ogur 2 1 Department of Mathematics, Faculty Art and Science, Ondokuz Mayis University, Samsun, Turkey 2 Department of Mathematics, Faculty Art and Science, Giresun University, Giresun, Turkey Received: 15 May 2017, Accepted: 1 June 2017 Published online: 9 April 2018. Abstract: The object of this paper is to introduce the double Ces` aro-Orlicz sequence space Ces (2) M using a Orlicz function M. Necessary and sufficient conditions under which the double Ces` aro-Orlicz sequence space Ces (2) M is nontrivial are presented. It is proved that double Ces` aro-Orlicz sequence spaces Ces (2) M are complete. Finally, it is obtained that if φ ∈ ∆ 2 (0) then the space Ces (2) M is separable and order continuous. Keywords: Double sequence, double Ces` aro-Orlicz sequence space, Luxemburg norm, Fatou property, order continuity. 1 Introduction As usual, N, R and R + denote the sets of positive integers, real numbers and nonnegative real numbers, respectively. A double sequence on a normed linear space X is a function x from N × N into X and briefly denoted by x =(x(i, j)). Throughout this work , w and w 2 denote the spaces of all single real sequences and double real sequences, respectively. First of all, let us recall preliminary definitions and notations. Definition 1. If for every ε > 0 there exists n ε ∈ N such that   x k,l − a   X < ε whenever k, l > n ε then a double sequence  x k,l  is said to be converge (in terms of Pringsheim) to a ∈ X [12]. A double sequence  x k,l  is called a Cauchy sequence if and only if for every ε > 0 there exists n 0 = n 0 (ε ) such that   x k,l − x p,q   < ε for all k, l , p, q ≥ n 0 . A double series is infinity sum ∑ ∞ k,l =1 x k,l and its convergence implies the convergence by ‖.‖ X of partial sums sequence {S n,m }, where S n,m = ∑ n k=1 ∑ m l =1 x k,l (see [1], [6]). Definition 2. If each double Cauchy sequence in X converges an element of X according to norm of X, then X is said to be a double complete space. A normed double complete space is said to be a double Banach space [1]. Definition 3. A Banach space (X , ‖.‖) which is a subspace of w (2) is said to be double K¨ othe sequence space if: (i) for any x ∈ w (2) and y ∈ X such that |x(i, j)|≤|y(i, j)| for all i, j ∈ N, we have x ∈ X and ‖x‖≤‖y‖, (ii) there is x ∈ X with x(i, j) = 0 for all i, j ∈ N. An element x from a double K¨ othe sequence space X is called order continuous if for any sequence (x n ) in X + (the positive cone of X ) such that x n ≤|x| for all n ∈ N and x n → 0 coordinatewise, we have ‖x n ‖→ 0. ∗ Corresponding author e-mail: oguz.ogur@giresun.edu.tr c  2018 BISKA Bilisim Technology