Topological Structures and Continuity: A Conceptual and Applied Study of Modern Topology

Abstract

Topology is among the most basic and the most interrelated fields in the contemporary mathematics with focus on the qualitative features of space that cannot be deformed and is the same under continuous deformations. Topology focuses more on concepts like continuity, connectedness, compactness, and convergence unlike geometry, which is concerned with matters like length and angle, which are metric properties. Topology has been developed as an abstract branch of mathematics into a fully developed and powerful discipline with major applications into analysis, physics, computer science, and data science in the last century. The study provides a conceptual and practical analysis of the existing topology, including its structures, important theoretical concepts, and methodologies. It covers topological spaces, continuous functions and basic properties like compactness and connectedness with a major focus on their importance in modern mathematics, and other trans-disciplinary use. The idea in this research is to determine the long-term relevance of topology as both a theoretical and practical field of study in mathematics through a combination of old-fashioned topology and new applications.