In this paper, we introduced and studied (1,2,3)-α-open set, (1,2,3)-α-neighborhood system, (1,2,3)-α-derived, (1,2,3)-α-closure, (1,2,3)-α-interior, (1,2,3)-α-exterior, (1,2,3)-α-boundary, (1,2,3)-α-convergence of nets, and (1,2,3)-α-convergence of filters in fuzzifying tritopological spaces.

Let G = (V, E) be a simple connected graph, then the basis number of G is denoted by b(G) and is defined by the least positive integer k such that the graph G has a k − fold basis for it is cycle space. In this paper we studied the basis number of Mycielski’s graph for some cog-special graphs, and we compute the basis number of Mycielski’s graph for cog-path graph, cog-cycle graph, cog-star graph, and cog-wheel graph.

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The present article introduce 0 (1,2,3) (Kolmogorov), 1 (1,2,3) (Fréchet), 2 (1,2,3) (Hausdorff), ℛ (1,2,3) (-regular), (1,2,3) (-normal), 0 (1,2,3) , 1 (1,2,3) and 2 (1,2,3) separation axioms in fuzzifying tri-topological spaces and studying the relation among them and also some of their properties.

In this paper, we define and study extensively a new type of external sets in R , we call it "convex galaxies". We show that these convex external sets may be classified in some definite types. More precisely, we obtain the following : (1) Let R G  be a convex galaxy which is symmetric with respect to zero, then