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Topological Spaces from Distance to Neighborhood

by Gerard Buskes

  • ISBN: 9780387949949
  • ISBN10: 0387949941

Topological Spaces from Distance to Neighborhood

by Gerard Buskes

  • List Price: $74.99
  • Binding: Hardcover
  • Publisher: Springer Verlag
  • Publish date: 07/01/1997
  • ISBN: 9780387949949
  • ISBN10: 0387949941
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Description: I The Line And The Plane.- 1 What Topology Is About.- Topological Equivalence.- Continuity and Convergence.- A Few Conventions.- Extra: Topological Diversions.- Exercises.- 2 Axioms for '.- Extra: Axiom Systems.- Exercises.- 3 Convergent Sequences and Continuity.- Subsequences.- Uniform Continuity.- The Plane.- Extra: Bolzano (1781-1848).- Exercises.- 4 Curves in the Plane.- Curves.- Homeomorphic Sets.- Brouwer's Theorem.- Extra: L.E.J. Brouwer (1881-1966).- II Metric Spaces.- 5 Metrics.- Extra: Camille Jordan (1838-1922).- Exercises.- 6 Open and Closed Sets.- Subsets of a Metric Space.- Collections of Sets.- Similar Metrics.- Interior and Closure.- The Empty Set.- Extra: Cantor (1845-1918).- Exercises.- 7 Completeness.- Extra: Meager Sets and the Mazur Game.- Exercises.- 8 Uniform Convergence.- Extra: Spaces of Continuous Functions.- Exercises.- 9 Sequential Compactness.- Extra: The p-adic Numbers.- Exercises.- 10 Convergent Nets.- Inadequacy of Sequences.- Convergent Nets.- Extra: Knots.- Exercises.- 11 Transition to Topology.- Generalized Convergence.- Topologies.- Extra: The Emergence of the Professional Mathematician.- Exercises.- III Topological Spaces.- 12 Topological Spaces.- Extra: Map Coloring.- Exercises.- 13 Compactness and the Hausdorff Property.- Compact Spaces.- Hausdorff Spaces.- Extra: Hausdorff and the Measure Problem.- Exercises.- 14 Products and Quotients.- Product Spaces.- Quotient Spaces.- Extra: Surfaces.- Exercises.- 15 The Hahn-Tietze-Tong-Urysohn Theorems.- Urysohn's Lemma.- Interpolation and Extension.- Extra: Nonstandard Mathematics.- Exercises.- 16 Connectedness.- Connected Spaces.- The Jordan Theorem.- Extra: Continuous Deformation of Curves.- Exercises.- 17 Tychonoff's Theorem.- Extra: The Axiom of Choice.- Exercises.- IV Postscript.- 18 A Smorgasbord for Further Study.- Countability Conditions.- Separation Conditions.- Compactness Conditions.- Compactifications.- Connectivity Conditions.- Extra: Dates from the History of General Topology.- Exercises.- 19 Countable Sets.- Extra: The Continuum Hypothesis.- A Farewell to the Reader.- Literature.- Index of Symbols.- Index of Terms.
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