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>

This article discusses the treatment of the income tax of second category generated from the sale of real property made by individuals. Its development will focus on the following points:1) The tax determination described sequentially and according to law; 2) The exceptionalcases where there is not capital gain on the sale of a real property; 3) The determination ofthe tax basis in the cases where the transferor acquired free of charge the property that willbe transferred; 4) The obligation of using means of payment accepted by the banking andfinancial system in the operation; 5) The opportunity to pay the tax; 6) The exchange rate tobe used when the operation is performed in foreign currency; 7) The resolution of everydaylife cases related to the tax calculation; and 8) The summary of the main points of this article. / El presente artículo versa sobre el tratamiento del Impuesto a la Renta de segunda categoría generado por la enajenación de inmuebles por parte de personas naturales. Su desarrollo gira en torno a los siguientes puntos: 1) El cálculo del Impuesto descrito de manera secuencial y según la base normativa; 2) Los supuestos excepcionales en los que no se considera que se ha generado ganancia de capital en la enajenación de un inmueble; 3) La determinación del costo computable en aquellos casos en que el enajenante adquirió de forma gratuita el inmueble que va a ser enajenado; 4) La obligación de bancarizar la operación; 5) La oportunidad de pago del impuesto; 6) El tipo de cambio a utilizar cuando la operación se realiza en moneda extranjera; 7) La resolución de casos de la vida cotidiana relacionados con el cálculo del impuesto; y, 8) El resumen de los puntos principales del artículo.