Topology is a silent inducer and a strong trend setter as it is a fundamental field in mathematics. It provides many basic concepts for modern analysis, hence many Mathematicians and Scientists apply the concept of Topology to understand the real world phenomena.
The three basic foundations in topology are general Topology, Algebraic Topology and Differential Topology. Grills, which is the main focus of this thesis comes under the head of general Topology. The idea of grills was introduced by Choquet in 1947. It is observed from the literature that the concept of grills is a powerful, supporting tool like nets and filters. B.Roy and M.N.Mukherjee developed the topology induced by grills. Further they proposed the definition of compactness through grills in and extended their study to fuzzy grill topology.
Fuzzy set was introduced by Zadeh. Fuzzy topology was initiated by Chang and it paved a way for a new era of fuzzy topology. Several researchers conducted on the generalizations of the notion of fuzzy topology. The intuitionistic fuzzy set was first published by K.Atanassov. Later topological structures in fuzzy topological spaces is generalized to “ Intuitionistic fuzzy topological spaces” by Coker in. Athar and Ahmad defined the notion of fuzzy boundary in FTS and studied the properties of fuzzy semi boundary. [...]
Inhaltsverzeichnis (Table of Contents)
- A PRELUDE TO THE STUDY
- Introduction
- Grill Topology
- Grill Compactness
- Intuitionistic C Fuzzy boundary
- NEW CLASSES OF SETS AND THEIR APPLICATIONS
- Grill generalized b-closed sets
- Ggfp-closed sets in fuzzy G-space
- Fuzzy normal and regular spaces by Ggfp-closed sets
- ÿ (X, t, G) a new class of sets
- G-COMPACTNESS, G-PARACOMPACTNESS AND G-WEAK COMPACTNESS
- G-0 compactness
- G-0 paracompactness
- Principalgrill [A] and its regularity
- Weak compactness by 0-open sets
- NEARLY COMPACTNESS AND CONVERGENCE OF GRILLS
- Grill near compactness
- Continuous mappings in nearly compact grill topological spaces
- 8-convergence and 8-adherence of grills
- Properties of ASμ-spaces
Zielsetzung und Themenschwerpunkte (Objectives and Key Themes)
This thesis delves into the investigation of weaker forms of compactness via grills, aiming to expand the understanding of topological spaces and their properties. The research examines various aspects of compactness, including its relationship to different types of sets and spaces, and investigates the convergence of grills within topological spaces.
- Introducing new classes of sets and their applications in topological spaces
- Exploring the concepts of G-compactness, G-paracompactness, and G-weak compactness
- Investigating the concept of grill near compactness and its connection to continuous mappings
- Analyzing the convergence and adherence of grills within topological spaces
- Examining the properties of specific types of spaces, such as ASμ-spaces, in relation to grill-based compactness
Zusammenfassung der Kapitel (Chapter Summaries)
The first chapter introduces the fundamental concepts of grill topology, grill compactness, and intuitionistic C fuzzy boundary, setting the stage for the subsequent investigation. Chapter two focuses on introducing new classes of sets, including grill generalized b-closed sets, Ggfp-closed sets in fuzzy G-space, and a novel class of sets denoted as ÿ (X, t, G). These sets are analyzed in relation to their properties and applications within topological spaces. Chapter three delves into the concepts of G-compactness, G-paracompactness, and G-weak compactness, exploring their relationships to various topological properties. Chapter four investigates the concept of grill near compactness, its connection to continuous mappings, and the convergence and adherence of grills within topological spaces. The chapter also examines the properties of ASμ-spaces in relation to grill-based compactness.
Schlüsselwörter (Keywords)
This thesis centers on the exploration of weaker forms of compactness within topological spaces using the framework of grills. Key concepts include grill topology, grill compactness, new classes of sets, G-compactness, G-paracompactness, G-weak compactness, grill near compactness, convergence of grills, and ASμ-spaces. The research contributes to the understanding of topological properties and their relationships to different types of sets and spaces.
- Quote paper
- Karthika Arumugam (Author), 2013, Investigation on weaker forms of compactness via grills, Munich, GRIN Verlag, https://www.grin.com/document/306767
