The Continuum

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.