Pairwise Compactness in Bi-Isotonic Spaces

COMPACTNESS IN BI-ISOTONIC SPACES SUMMARY This thesis study consists of five sections aiming to introduce the pairwise compactness of bi-isotonic spaces and investigate their properties. The sections explaining key concepts provide essential information on bi-isotonic spaces, bitopological spaces, and pairwise compactness in bitopological spaces by summarizing the findings of previous research on the related subjects. The first section presents a detailed literature review related to the thesis topic. The objective of this section is to extensively discuss previous studies and significant findings regarding bi-isotonic spaces and the theory of compactness. The literature review aims to provide the readers with a broad perspective by presenting the foundational reference points that constitute the basis of the thesis. In this context, the pioneering studies related to the bi-isotonic and bitopological spaces and the significant results achieved are thoroughly expressed. The presentation of the literature aims to deepen the existing knowledge, emphasize the context and importance of the research questions and objectives of the thesis, and highlight the innovative contribution of the thesis in this field. The first section enables readers to understand the scope of the thesis, identify gaps in the literature, and recognize the thesis's endeavor to fill these gaps. Thus, it provides a better context for the analysis and conclusions to be presented in the subsequent sections. The second section is a comprehensive chapter that examines the fundamental definitions and properties of a bitopological space which was defined by Kelly in 1962, such as a triple where and are topologies on a non-empty set . This section offers information for in-depth insights into bitopological spaces and contributes to understanding more advanced concepts in bi-isotonic spaces. Bitopological spaces are a class derived from general topological spaces, possessing additional structure and properties. In this section, the basic definitions of bitopological spaces are taken as a starting point. These definitions show that bitopological spaces give a more effective way to study the relationships and properties of points in space by considering two different topological structures, thus open sets or neighborhoods. This section also elaborates on how mappings between bitopological spaces are defined, their impact on their continuity properties, and the generalizations of separation axioms in bitopological spaces. To provide a solid foundation on bi-isotonic spaces, this section thoroughly examines the relationships between bitopological spaces and topological spaces, as well as the comparative analysis of different classes of bitopological spaces, which are also included in this section. The third section provides a detailed explanation of fundamental concepts and theorems concerning the bi-isotonic spaces and subspaces, continuity, and separation axioms in these spaces. Bi-isotonic spaces are particular spaces defined as follows: A generalized bi-closure space is called bi-isotonic space where and defined on the power set of a non-empty set are the closure operators satisfying only the grounded and isotony axioms. In topological spaces, a closure operator is well-known as a mathematical operation satisfying each Kuratowki axiom. However, if an operator satisfies only the grounded axiom and isotony axiom for all it is called isotony operator. In that way, in the bi-isotonic spaces the operators , and are isotonic. This allows us to construct two structures on a space generalizing bitopological spaces by these isotony operators. This relatively new definition of bi-isotonic space in the realm of topology leads to define more general concepts in the area. As can be seen from this definition, bi-isotonic spaces exhibit differences in density among their subsets. This concept allows for the mathematical investigation of the existence of subsets with different densities in a space. This section also covers fundamental concepts and theorems related to subspaces, continuity, and separation axioms in bi-isotonic spaces. Subspaces refer to the condition where subsets of a bi-isotonic space also form bi-isotonic spaces. Bi-continuity signifies the specific continuity properties of transformations in a bi-isotonic space. Separation axioms imply the properties of bi-isotonic spaces that separate points via their neighborhoods. Consequently, the definition of bi-isotonic spaces, their specific properties, and explanations through examples constitute a stage for the original contributions and serve the main objective of the thesis by emphasizing the importance of bi-isotonic spaces in mathematical analysis. The foundational information presented in this section facilitates the comprehension of the analysis and conclusions to be presented in the subsequent sections, contributing to the narrative structure of the thesis. The fourth section constitutes the most significant and original part of the thesis based on the foundational information provided in previous sections on bitopological and bi-isotonic spaces. Three different pairwise compactness definitions are given such as pairwise S-compactness referring to the study of Swart (1971); pairwise B-compactness referring to the study of Birsan (1969); and pairwise FHP- compactness based on the study of Fletcher, Hoyle ve Patty (1969). From these three definitions, it is seen that if a bi-isotonic space is pairwise S-compact pairwise B-compact pairwise FHP- compact. In that regard, pairwise S-compact bi-isotonic space is studied, and it is called only pairwise compact bi-isotonic space in the rest of the thesis. Then new definitions and theorems are stated on pairwise compact subsets of bi-isotonic space. Also, the relationship between pairwise compactness and pairwise Hausdorff spaces is investigated. However, a new situation occurs that the relationship between pairwise compactness and pairwise Hausdorff spaces in bitopological spaces does not exist in bi-isotonic spaces. It is proved that the space has to be a biclosure space. Additionally, the relationship between pairwise compactness and limit points in bi-isotonic spaces is studied. Related theorems are expressed and proved. Finally, it is found whether the pairwise compactness is preserved under bi-continuity or not. Thus, it is seen that pairwise compactness is a topological property in bi-isotonic spaces. These main results in this section make a valuable contribution to the literature by providing a new perspective on the compactness theory. A more detailed understanding of the compactness properties of bi-isotonic spaces paves the way for further research and advancements in the field of mathematical analysis. This thesis study aims to provide a significant contribution to the pairwise compactness of bi-isotonic spaces, thereby supporting the advancement of research in this area within mathematical analysis. By improving the understanding of this particular type of space, this study will contribute to the general knowledge of mathematical analysis. The obtained results will represent a crucial step towards a more in-depth examination of the compactness properties of bi-isotonic spaces, leading to discoveries and advancements in both theoretical and applied mathematics.