ABSTRACT
The idea of complete metric spaces although rather straightforward, brings forth some very elegant results. The fact that we can embed any space in a complete metric space gives us all the more reason to study it. The work of Rene Baire on such spaces form a basic framework upon which a lot of modern analysis is based upon with far-reaching applications to Functional Analysis, non-linear analysis and even theory of several complex variables.
In this chapter, we first look at sequences and the motivation behind their definition. The definition of complete spaces follows along with various standard examples, certain techniques that can be used to test whether a given space is complete and a constructive proof to completion of any given metric space. We then move on to the main result of the section viz. Baire Category Theorem with two non-constructive proof and its various applications that illustrate the usefulness of this tool.
