On Soft Topological Spaces

Abstract

Chapter 1 is the introductory chapter. It contains also the basic concepts and properties of topological spaces such as neighborhoods, closure, interior and separation axioms. The basic concepts and properties of the double topological space (DT S, for short) are presented. Further, this chapter contains the basic notions related to soft sets and soft topological spaces. Also, in this chapter we illustrate that the sufficiency of the Theorem 9.2.11 [30], is incorrect by giving a counter Example and we show that, in [25] Remark 4.2, Example 4.3, Theorem 4.6 and Example 4.15 are not true, in general.

In Chapter 2, we study some topological properties of double topological spaces (DT −spaces, for short), also we introduce some generalized of double separation axioms of DT −spaces based on double separation axioms [20]. Moreover, we in- troduce some types of double connected spaces (D-connected spaces, for short) such as q-double connected (qD-connected, for short), double C1−connected (DC1 −connected, for short), strongly (q-)double connected (strongly (q)D-connected, for short), double hyperconnected (D-hyperconnected, for short), q-double hy- perconnected (qD-hyperconnected, for short), double component (D-component, for short) and q-double component (qD-component, for short). Some examples are given to illustrate this notion. In addition, double T 2 −space presented in