Anti-spaces of strongly k-spaces and related maps

This thesis consists of five chapters. The first chapter is devoted to detailed literature knowledge related to subject of the thesis. In the second chapter, the definitions of pre-open (pre-closed) set, supratopological space, pre-interior, pre-clousure, pre-accumulation point, pre-neighborhood of a set and pre-irresolute, pre-continuous, pre-resolute, strongly pre-continuous functions, strongly compact set and strongly compact space are introduced. In the third chapter, the concepts of k-space and anti-space are introduced and the theorems about anti-spaces are stated and proved. Subsequently the results of these theorems are given. Later, the necessary and sufficient conditions for a space to be a k-space are explained. Also, the definition of KC space is given. The function f*:X*->Y*, which is corresponding to the function f:X->Y , is introduced where X and Y are topological spaces and X* and Y* are, respectively, anti-spaces of them. Moreover, the relationships between these maps are discussed. The fourth chapter is the original part of this study. At the beginning of this chapter the anti-space of a space given with a supratopology constituted by pre-open sets is introduced. The definition of pKC space is given and the relationships between the strongly compact sets and pre-closed sets in pKC spaces which are strongly k-spaces are examined. The functions between the spaces given with supratopologies constituted by pre-open sets are defined. By studying the functions between pKC spaces which are strongly k-spaces and the corresponding functions between the anti-spaces of them the related theorems are expressed and proved. By investigating the relations between these functions the obtained results are summarized by the aid of a table. In the fifth chapter of this thesis, a brief summary of this study is given and some suggestions are proposed for new investigations.