This text explains nontrivial applications of metric space topology to analysis. Covers metric space, point-set topology, and algebraic topology. Includes exercises, selected answers, and 51 illustrations. 1983 edition.
This text explains nontrivial applications of metric space topology to analysis. Covers metric space, point-set topology, and algebraic topology. Includes exercises, selected answers, and 51 illustrations. 1983 edition.
ONE METRIC SPACES 1 Open and closed sets 2 Completeness 3 The real line 4 Products of metric spaces 5 Compactness 6 Continuous functions 7 Normed linear spaces 8 The contraction principle 9 The Frechet derivative TWO TOPOLOGICAL SPACES 1 Topological spaces 2 Subspaces 3 Continuous functions 4 Base for a topology 5 Separation axioms 6 Compactness 7 Locally compact spaces 8 Connectedness 9 Path connectedness 10 Finite product spaces 11 Set theory and Zorn's lemma 12 Infinite product spaces 13 Quotient spaces THREE HOMOTOPY THEORY 1 Groups 2 Homotopic paths 3 The fundamental group 4 Induced homomorphisms 5 Covering spaces 6 Some applications of the index 7 Homotopic maps 8 Maps into the punctured plane 9 Vector fields 10 The Jordan Curve Theorem FOUR HIGHER DIMENSIONAL HOMOTOPY 1 Higher homotopy groups 2 Noncontractibility of Sn 3 Simplexes and barycentric subdivision 4 Approximation by piecewise linear maps 5 Degrees of maps BIBLIOGRAPHY LIST OF NOTATIONS SOLUTIONS TO SELECTED EXERCISES INDEX
ONE METRIC SPACES 1 Open and closed sets 2 Completeness 3 The real line 4 Products of metric spaces 5 Compactness 6 Continuous functions 7 Normed linear spaces 8 The contraction principle 9 The Frechet derivative TWO TOPOLOGICAL SPACES 1 Topological spaces 2 Subspaces 3 Continuous functions 4 Base for a topology 5 Separation axioms 6 Compactness 7 Locally compact spaces 8 Connectedness 9 Path connectedness 10 Finite product spaces 11 Set theory and Zorn's lemma 12 Infinite product spaces 13 Quotient spaces THREE HOMOTOPY THEORY 1 Groups 2 Homotopic paths 3 The fundamental group 4 Induced homomorphisms 5 Covering spaces 6 Some applications of the index 7 Homotopic maps 8 Maps into the punctured plane 9 Vector fields 10 The Jordan Curve Theorem FOUR HIGHER DIMENSIONAL HOMOTOPY 1 Higher homotopy groups 2 Noncontractibility of Sn 3 Simplexes and barycentric subdivision 4 Approximation by piecewise linear maps 5 Degrees of maps BIBLIOGRAPHY LIST OF NOTATIONS SOLUTIONS TO SELECTED EXERCISES INDEX
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