Um estudo introdutório à topologia

Abstract

This work aims to present some fundamental concepts of Topology, highlighting its importance and applications, acquiring knowledge in the field of topology, thus guiding future studies in the area, as well as beginning to address a gap encountered during my undergraduate degree in Mathematics. The approach begins with a historical motivation, featuring classical problems such as the Seven Bridges of Königsberg and Euler's polyhedron formula, which marked the emergence of topological thinking. Next, essential preliminary notions are reviewed, including sets, mappings, functions, real numbers, and vector spaces. With this foundation, the concept of metric spaces is introduced, covering definitions, open and closed balls, and spheres. From there, the formal definition of a topology on a set is presented, along with the concepts of open and closed sets, accompanied by examples. The text continues with the study of continuity and homeomorphisms, which are central to understanding equivalences between topological spaces under continuous deformations. As a practical complement, an appendix includes an interactive model developed in GeoGebra, illustrating the continuous deformation of a cube into a sphere, providing a visual example of topological equivalence. As an introductory study, this work does not intend to exhaust all related topics and recommends that readers consult the indicated references for further depth