The Continuum - Rudolf Taschner

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Résumé :
Das Buch ist gut geeignet als Textgrundlage f?r Seminare aus reeller Analysis (vorzugsweise f?r Seminare, nachdem die Studierenden den ?blichen formal-axiomatischen Zugang zu den reellen Zahlen erfahren haben). In this small text the basic theory of the continuum, including the elements of metric space theory and continuity is developed within the system of intuitionistic mathematics in the sense of L.E.J. Brouwer and H. Weyl. The main features are proofs of the famous theorems of Brouwer concerning the continuity of all functions that are defined on whole intervals, the uniform continuity of all functions that are defined on compact intervals, and the uniform convergence of all pointwise converging sequences of functions defined on compact intervals. The constructive approach is interesting both in itself and as a contrast to, for example, the formal axiomatic one....

Biographie:
Rudolf Taschner is Professor of Mathematics at the Institute for Analysis and Scientific Computing, Technical University Vienna, Austria. In his recent book Der Zahlen gigantische Schatten (Vieweg 2004) he describes how intensively numbers penetrate the aspects of our life, and how far the shadows of numbers reach....

Sommaire:
1 Introduction and historical remarks.- 1.1 Farey fractions.- 1.2 The pentagram.- 1.3 Continued fractions.- 1.4 Special square roots.- 1.5 Dedekind cuts.- 1.6 Weyl?s alternative.- 1.7 Brouwer?s alternative.- 1.8 Integration in traditional and in intuitionistic framework.- 1.9 The wager.- 1.10 How to read the following pages.- 2 Real numbers.- 2.1 Definition of real numbers.- 2.1.1 Decimal numbers.- 2.1.2 Rounding of decimal numbers.- 2.1.3 Definition and examples of real numbers.- 2.1.4 Differences and absolute differences.- 2.2 Order relations.- 2.2.1 Definitions and criteria.- 2.2.2 Properties of the order relations.- 2.2.3 Order relations and differences.- 2.2.4 Order relations and absolute differences.- 2.2.5 Triangle inequalities.- 2.2.6 Interpolation and Dichotomy.- 2.3 Equality and apartness.- 2.3.1 Definition and criteria.- 2.3.2 Properties of equality and apartness.- 2.4 Convergent sequences of real numbers.- 2.4.1 The limit of convergent sequences.- 2.4.2 Limit and order.- 2.4.3 Limit and differences.- 2.4.4 The convergence criterion.- 3 Metric spaces.- 3.1 Metric spaces and complete metric spaces.- 3.1.1 Definition of metric spaces.- 3.1.2 Fundamental sequences.- 3.1.3 Limit points.- 3.1.4 Apartness and equality of limit points.- 3.1.5 Sequences in metric spaces.- 3.1.6 Complete metric spaces.- 3.1.7 Rounded and sufficient approximations.- 3.2 Compact metric spaces.- 3.2.1 Bounded and totally bounded sequences.- 3.2.2 Located sequences.- 3.2.3 The infimum.- 3.2.4 The hypothesis of Dedekind and Cantor.- 3.2.5 Bounded, totally bounded, and located sets.- 3.2.6 Separable and compact spaces.- 3.2.7 Bars.- 3.2.8 Bars and compact spaces.- 3.3 Topological concepts.- 3.3.1 The cover of a set.- 3.3.2 The distance between a point and a set.- 3.3.3 The neighborhood of a point.- 3.3.4 Dense and nowhere dense.- 3.3.5 Connectedness.- 3.4 The s-dimensional continuum.- 3.4.1 Metrics in the s-dimensional space.- 3.4.2 The completion of the s-dimensional space.- 3.4.3 Cells, rays, and linear subspaces.- 3.4.4 Totally bounded sets in the s-dimensional continuum.- 3.4.5 The supremum and the infimum.- 3.4.6 Compact intervals.- 4 Continuous functions.- 4.1 Pointwise continuity.- 4.1.1 The concept of function.- 4.1.2 The continuity of a function at a point.- 4.1.3 Three properties of continuity.- 4.1.4 Continuity at inner points.- 4.2 Uniform continuity.- 4.2.1 Pointwise and uniform continuity.- 4.2.2 Uniform continuity and totally boundness.- 4.2.3 Uniform continuity and connectedness.- 4.2.4 Uniform continuity on compact spaces.- 4.3 Elementary calculations in the continuum.- 4.3.1 Continuity of addition and multiplication.- 4.3.2 Continuity of the absolute value.- 4.3.3 Continuity of division.- 4.3.4 Inverse functions.- 4.4 Sequences and sets of continuous functions.- 4.4.1 Pointwise and uniform convergence.- 4.4.2 Sequences of functions defined on compact spaces.- 4.4.3 Spaces of functions defined on compact spaces.- 4.4.4 Compact spaces of functions.- 5 Literature.