This paper examines the relationship between Borel sets and Baire functions.

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>

No description available.

O principal objetivo desse trabalho é o estudo da questão da existência de isomorfismos entre as classes de Baire sobre [0,1]. Para isso, desenvolvemos os principais resultados concernentes às relações entre as classes de Baire sobre [0,1]. A saber: (1) As classes de Baire são isométricas como álgebras de Banach a espaços da forma C(K); (2) As classes de Baire são subespaços próprios umas das outras, até o primeiro ordinal não enumerável, onde elas estabilizam; (3) As classes de Baire não são subespaços complementados umas das outras; (4) As classes de Baire não são isométricas umas às outras como espaços de Banach. Por fim, apresentamos as respostas conhecidas para a questão isomórfica, sendo que para tal, utilizamos os resultados mencionados acima. / The main purpose of this work is the study of the question about the existence of isomorphisms between the Baire classes on [0,1]. In order to do that, we develop the most important results concerning the relations between the Baire classes on [0,1]. Those results are: (1) The Baire classes are isometric as Banach algebras to spaces of the form C(K); (2) The Baire classes are proper subspaces each one of the others, until the first uncountable ordinal, when they stabilise; (3) The Baire classes aren\'t complemented subspaces each one of the others; (4) There aren\'t linear isometries between the Baire classes. Finally we presente the known answers to the isomorphic question, using for this the results mentioned above.

In the study of Banach spaces, the development of some key properties require studying topologies on the collection of closed convex subsets of the space. The subcollection of closed linear subspaces is studied under the relative slice topology, as well as a class of topologies similar thereto. It is shown that the collection of closed linear subspaces under the slice topology is homeomorphic to the collection of their respective intersections with the closed unit ball, under the natural mapping. It is further shown that this collection under any topology in the aforementioned class of similar topologies is a strong Choquet space. Finally, a collection of category results are developed since strong Choquet spaces are also Baire spaces.

This paper is dedicated to the beginning of the development of a book introducing topological games and selection principles. Here, the classical games (such as the Banach-Mazur) and selection principles (such as the Rothberger or Menger properties) are presented. The most notable applications are also displayed both the classical (such as the characterization of Baire spaces with the Banach-Mazur game) and the recent (such as the relation between the Menger property and D-spaces). In addition to the content for the book, a problem in finite combinatorics that was found in the study of positional strategies is presented (as well as a partial solution) together with some results regarding new variations of classical selection principles and games, which give rise to the characterization of some notable spaces. / Este trabalho é dedicado ao início do desenvolvimento de um livro introdutório à jogos topológicos e princípios seletivos. Aqui, são apresentados os clássicos jogos (tais como o de Banach-Mazur) e princípios seletivos (tais como a propriedade de Rothberger ou de Menger). Também são exibidas as aplicações mais notáveis encontradas na literatura tanto as mais tradicionais (tais como a caracterização de espaços de Baire com o jogo de Banach-Mazur), como as mais atuais (tais como a relação entre a propriedade de Menger e D-espaços). Além do conteúdo voltado para o livro, são apresentados um problema de combinatória finita (assim como uma solução parcial para tal) que foi encontrado com o estudo de estratégias posicionais e alguns resultados envolvendo novas variações de princípios de seleção e jogos clássicos, possibilitando a caracterização de alguns espaços notáveis.

This thesis consists of five research papers. The first paper: We prove that under certain conditions, the existence of an isomorphism between spaces of continuous affine functions on the compact convex sets imposes home- omorphism between the sets of its extreme points. The second: We investigate a transfer of descriptive properties of elements of biduals of Banach spaces con- strued as functions on dual unit balls. We also prove results on the relation of Baire classes and intrinsic Baire classes of L1-preduals. The third: We identify intrinsic Baire classes of X with the spaces of odd or homogeneous Baire functions on ext BX∗ , provided X is a separable real or complex L1-predual with the set of extreme points of its dual unit ball of type Fσ. We also provide an example of a separable C∗ -algebra such that the second and second intrinsic Baire class of its bidual differ. The fourth: We generalize some of the above mentioned results for real non-separable L1-preduals. The fifth: We compute the distance of a general mapping to the family of mappings of the first resolvable class via the quantity frag and we introduce and investigate a class of mappings of countable oscillation rank.

L'analyse multifractale est l'étude des propriétés locales des ensembles de mesures ou de fonctions. Son importance est apparue dans le cadre de la turbulence pleinement développée. Dans ce cadre, l'expérimentateur n'a pas accès à la vitesse en tout point d'un fluide mais il peut mesurer sa valeur en un point en fonction du temps. On ne mesure donc pas directement la fonction vitesse du fluide, mais sa trace. Cette thèse sera essentiellement consacrée à l'étude du comportement local de traces de fonctions d'espaces de Besov : nous déterminerons la dimension de Hausdorff des ensembles de points ayant un exposant de Hölder donné (spectre multifractal). Afin de caractériser facilement l'exposant de Hölder et l'appartenance à un espace de Besov, on utilisera la décomposition de fonctions sur les bases d'ondelettes.Nous n'obtiendrons pas la valeur du spectre de la trace de toute fonction d'un espace de Besov mais sa valeur pour un ensemble générique de fonctions. On fera alors appel à deux notions de généricité différentes : la prévalence et la généricité au sens de Baire. Ces notions ne coïncident pas toujours, mais, ici on obtiendra les mêmes résultats. Dans la dernière partie, afin de déterminer la forme que peut prend un spectre multifractal, on construira une fonction qui est son propre spectre / Multifractal analysis consists in the study of local properties of set of measures or functions. Its importance appeared in the frame of fully developed turbulence. In this area, physicists do not know the velocity of a fluid at all points but they can measure its value in one point in function of time. Hence, they do not measure the velocity function of the fluid but its trace.This thesis will be mainly dedicated to the study of local behavior of traces of Besov functions: we will determine the Hausdorff dimension of sets of points with a given Hölder exponent (the so-called multifractal spectrum). In order to easily characterize Hölder exponent and Besov spaces, we will use wavelet decomposition. We will not get the value of the multifractal spectrum of the trace of all functions of a Besov space, but its value for a generic set of functions. Then, we will use two notions of genericity : prevalence and Baire's genericity. Even if generic and prevalent properties can be different, here they will be the same.In the last part, in order to establish what a multifractal spectrum shape can be, we will construct a function which is its own spectrum

Working within the Zermelo-Frankel Axioms of set theory, we will introduce two important contradictory axioms: Axiom of Choice and Axiom of Determinacy. We will explore perfect polish spaces and games on these spaces to see that the Axiom of Determinacy is inconsistent with the Axiom of Choice. We will see some of the major consequences of accepting the Axiom of Determinacy and how some of these results change when accepting the Axiom of Choice. We will consider 2-player games of perfect information wherein we will see some powerful results having to do with properties of the real numbers. We will use a game to illustrate a weak proof of the continuum hypothesis.