Regular local functions in ideal topological spaces

The importance of ideal topological spaces is well known. Due to its importance, ideal topological spaces are discussed in the current literature. Recently many published works made on local function used in ideal topological spaces can be found in related literature. "Semi Local Functions in Ideal Topological Spaces", "Closure Local Functions", and " (┤)^p and ψ_p-Operator" can be mentioned among such works those aim to define such functions. In general, the researchers prefer using the generalized open sets instead of topology in ideal topological spaces. Obtaining a Kuratowski closure operator with the help of local functions is an important detail in ideal topological space. However, it is not possible to obtain a Kuratowski closure operator from many of these local functions proposed by the above mentioned works. In order to address the lack of such an operator, the main goal of this thesis is to introduce another local function to give possibility of obtaining a Kuratowski closure operator. On the other hand, regular local functions defined for ideal topological spaces have not been found in the current literature. Therefore, again to address the lack of such a function, regular local functions for the ideal topological spaces has been described within this thesis. This is the second goal of the thesis. Moreover, with the help of regular local functions Kuratowski closure operators k_I^(*d) and τ^(*d) topology are obtained. Many theorems in the literature have been revised according to the definition of regular local functions. The revised new theorems and other derived results are also included in this thesis. With the respect of above mentioned goals first, the fundamentals of the subject are presented in the thesis. Later, the regular open sets and semi open sets those often needed throughout the study are defined to create a base for deriving regular local functions. Local functions, semi-local functions and regular local functions are defined in ideal topological spaces and their interrelations are compared. Then, the new conditions which provided by ψ_d-operators defined with the help of regular local functions are presented. Additionally, a new topology extracted from ψ_d-operator is given. The regular compatibility between the topology and ideality in the ideal topological space are also included by this thesis together with the obtained results presented under the title namely " ψ_d-C sets definition" as the research findings. This thesis consists of five chapters. After an introduction, a review on the published works available in current literature is presented. The works are summarized and briefly discussed in this chapter. Definition of semi-open sets and regular open sets, ideal topological spaces, local functions and related theorems are given in Chapter Three. The conditions of regular local functions which are extracted from definition of local functions in topological spaces can be found in the next chapter. Chapter Four includes comparison of regular local functions with the other local functions. A new closure operator and a new topology have obtained with the help of regular local functions in the same chapter. Finally, the study is discussed and concluded in the last part of the thesis.