A compacidade em alguns universos topológicos

Abstract

This work aims to study and establish relationships between compact sets and topology, emphasizing the compactness of the unitary closed ball in different contexts. For this, initially we dealt with topological spaces in Chapter 1, we developed basic concepts and tools that will be useful until we get to the central theme of compactness. Then, in Chapter 2, we focus the study on the more particular environment of metric spaces, where we develop the concepts and results in order to end the chapter with a compactness characterization of the unitary closed ball of space. Finally, in Chapter 3, we study how weak and weak* topologies, drawing as a final result the famous Banach- Alaoglu-Bourbaki Theorem, which tells us that the unitary closed ball in the topological dual of a space of Banach is weak* compact.