Topologías generalizadas y sus aplicaciones

Abstract

This work presents a study on generalized topological spaces, an extension of the theory of classical topological spaces introduced by Ákos Császár in the 20th century. These generalized spaces allow for a more flexible structure through a family of subsets known as generalized topology, which are utilized to analyze key properties such as compactness, connectedness, and separation conditions. In this context, category theory is introduced as a formal framework for relating these spaces through continuous morphisms and commutative diagrams, exploring concepts such as products, equalizers, and pullbacks within the category of generalized topological spaces. Additionally, applications of this structure are presented in areas such as graph theory and databases, highlighting its potential for the organization and classification of complex structures. The methodology includes formalizing definitions, proving theorems, and providing illustrative examples that emphasize the properties of generalized topological spaces and their applications in both pure mathematics and data analysis.