Introductory Problem Courses in Analysis and Topology

Front Matter....Pages i-vii
Front Matter....Pages ix-ix
Notations....Pages 1-1
The Real Numbers, Regarded as an Ordered Field....Pages 2-5
Functions, Limits, and Continuity....Pages 6-10
Integers. Sequences. The Induction Principle....Pages 11-14
The Continuity of IR....Pages 15-21
The Riemann Integral of a Bounded Function....Pages 22-27
Necessary and Sufficient Conditions for Integrability....Pages 28-31
Invertible Functions. Arc-length and Path-length....Pages 32-36
Point-wise Convergence and Uniform Convergence....Pages 37-39
Infinite Series....Pages 40-43
Absolute Convergence. Rearrangements of Series....Pages 44-45
Power Series....Pages 46-48
Power Series for Elementary Functions....Pages 49-50
Front Matter....Pages 51-51
Sets and Functions....Pages 53-56
Metric Spaces....Pages 57-59
Neighborhood Spaces and Topological Spaces....Pages 60-64
Cardinality. Finite and Countable Sets....Pages 65-68
The Completeness of IR. Uncountable Sets....Pages 69-70
The Schröder-Bernstein Theorem....Pages 71-71
Compactness in IR n ....Pages 72-75
Front Matter....Pages 51-51
Compactness in Abstract Spaces....Pages 76-77
The Use of Choice in Existence Proofs....Pages 78-80
Linearly Ordered Spaces....Pages 81-82
Mappings Between Metric Spaces....Pages 83-85
Mappings Between Topological Spaces....Pages 86-87
Connectivity....Pages 88-90
Well-ordering....Pages 91-92
The Existence of Well-orderings. Zorn’s Lemma....Pages 93-94
Back Matter....Pages 95-95