O teorema de Baire.

Abstract

In this paper we conducted a study on Metric Spaces presenting its definition along with some examples and results, looking for enough tools to demonstrate the theorem from Baire. This theorem that has applications in both Functional Analysis, how much in topology, which highlights its importance. In order to demonstrate this, we have done the study in the theory of spaces Metrics as: balls and spheres, limited sets, distance between point and set and between sets. Besides these, we study sequences, as well as the topology of metric spaces, open sets and closed sets. We also study the continuous functions, destacando a definição de homeomorfismo e de continuidade uniforme. And so we can analyze the theorem Baire’s investigation investigated the complete metric spaces and their relationships with the sequences of Cauchy. However, before presenting the theorem itself and demonstrating it, we made a brief biography from René-Louis Baire, and after that, we realize the generalization of the Interval Principle Fits in Analysis, which is an important basic result for the theory.