Punctual Graph Topological Space: Expansible and Non-Expansible Cases

The relationship between graph theory and topological space is a multifaceted one, where graph theory can be used to represent topological spaces, facilitating their understanding and the solution of associated problems in an effective manner. Additionally, graphs can be generated from topological spaces, opening up new avenues for research in various fields. In this research, we will explore the relationship between graph theory and topological space, and seek to develop a theoretical framework that combines graph topological space with graph theory, focusing on its practical applications in various fields. The main objective of this paper is to present a new method for developing and applying graph topological space in various fields, including urban planning and neuroscience, which can contribute to improving the quality of life in modern societies. The results obtained indicate that this work can contribute to a better understanding of life models and provide assistance in solving some problems in daily life, opening up new avenues for research in various fields, and creating new doors for innovation and improvement in areas of life