In this paper we studied the translation surfaces according to a new frame called q-frame in three dimensional Euclidean space. The curvatures of the translation surface are obtained in terms of q-frame curvatures. Finally some special cases are investigated for these surfaces.

We extend the path lifting property in homotopy theory for topological spaces to bitopological semigroups and we show and prove its role in the C(ℵ)-fibration property. We give and prove the relationship between the C(ℵ)-fibration property and an approximate fibration property. Furthermore, we study the pullback maps for C(ℵ)-fibrations.

The aim of this paper is to introduce the concepts of τ1τ2-regular generalized star star closed sets , τ1τ2-regular generalized star star open sets and study their basic properties in bitopological spaces. Keywords—τ1τ2-regular closed sets; τ1τ2-regular open sets; τ1τ2-regular generalized closed sets; τ1τ2-regular generalized star closed sets; τ1τ2-regular generalized star star closed sets.

In this paper we define $q$-spherical surfaces as the that contain absolute conic of Euclidean space a $q-$fold curve. Particular attention is paid to with singular points highest order. Two classes such surfaces, one and two $n-$fold points, are discussed in detail. We study their properties, give algebraic equations visualize them program {\it Mathematica}.

Considering a countable number of topologies on a set X, we introduce the notion of (א0)topological spaces as a generalization of the notions of both bitopological spaces and (ω)topological spaces, and study some of their properties. AMS Mathematics Subject Classification (2000): 54A10