Topology of soft metric spaces

In the second chapter in this thesis, a short literature information about the soft sets and some fundemantal concepts and properties are given. The soft metric spaces via soft elements are discussed in detail in the third chapter. The soft metric structure that established by the soft element on the soft sets is given, relations between the soft metric spaces and the classical metric spaces are examined and some examples are given in this chapter. Also, soft subspace, the consepts of distance and diameter and their some properties in soft metric spaces are given. Soft open balls, soft closed balls, soft neighboard, soft interior, soft open and soft closed sets are examined. Although the elemanter soft unions of soft open sets are soft open, the elemantary intersection of two soft sets is not open. So, the soft metric spaces are restricted (the condition M5) for the elemantary intersection of two soft sets to be open. It is seen that every metric space satifyng the condition (M5) is a elemantary soft topological space. It is worked on limit elements and closure elements as the concepts of topological in the elemantery soft metric topological space. In addition, the concepts as convergence of sequences of the soft elements, Cauchy sequences and completeness in soft metric spaces are given. In the fourth chapter, the compactness of soft metric spaces is considered. In this chapter, the concepts of sequentially closeness, sequentially compactness, soft net, soft totally boundedness, Lebesques element, soft covering, compactness, soft sequentially continuity and continuity are defined, and their basic properties are proved in soft metric spaces. Since every soft metric spaces is not a elementary soft topological space, it is seen that sequentially compactness is different from compactness. Nevertheless, it is proved that every sequentially compact soft set is compact, and every sequentially continuous mapping is continuous in the soft metric space satisfying (M5). It is also proved that some properties of sequentially compactness and compactness in this chapter. The results obtained in the scope of this thesis will be the basis for further studies in this context. Keywords: Soft set, soft element, elementary operations, soft metric, elementary soft topology, completeness, Banach fixed point theorem, totally boundedness, compactness, sequentially compactness, continuity.