In this thesis we give an exposition of the notion of category and the Baire category theorem as a set theoretical method for proving existence. The category method was introduced by René Baire to describe the functions that can be represented by a limit of a sequence of continuous real functions. Baire used the term functions of the first class to denote these functions. The usage of the Baire category theorem and the category method will be illustrated by some of its numerous applications in real and functional analysis. Since the usefulness, and generality, of the category method becomes fully apparent in Banach spaces, the applications provided have been restricted to these spaces. To some extent, basic concepts of metric topology will be revised, as the Baire category theorem is formulated and proved by these concepts. In addition to the Baire category theorem, we will give proof of equivalence between different versions of the theorem. Explicit examples, of first class functions will be presented, and we shall state a theorem, due to Baire, providing a necessary condition on the set of points of continuity for any function of the first class.

<p>In this thesis we give an exposition of the notion of <em>category </em>and the <em>Baire category theorem </em>as a set theoretical method for proving existence. The category method was introduced by René Baire to describe the functions that can be represented by a limit of a sequence of continuous real functions. Baire used the term <em>functions of the first class </em>to denote these functions.</p><p>The usage of the Baire category theorem and the category method will be illustrated by some of its numerous applications in real and functional analysis. Since the usefulness, and generality, of the category method becomes fully apparent in Banach spaces, the applications provided have been restricted to these spaces.</p><p>To some extent, basic concepts of metric topology will be revised, as the Baire category theorem is formulated and proved by these concepts. In addition to the Baire category theorem, we will give proof of equivalence between different versions of the theorem.</p><p>Explicit examples, of first class functions will be presented, and we shall state a theorem, due to Baire, providing a necessary condition on the set of points of continuity for any function of the first class.</p><p> </p>