Topics Covered

• Preliminaries
  • Sets, Subsets & Complements, Union & Intersection
  • Relations and Functions
  • Orderings, Order-Complete Sets, Chains, Extension of Order-Preserving Functions
  • The Real Numbers and Countable Sets
  • Cardinal & Ordinal Numbers
  • Cartesian Products
  • The Hausdorff Maximal Principle
• Topological Spaces
  • Topologies & Neighborhoods
  • Closed Sets
  • Accumulation Points
  • Closure
  • Interior & Boundary
  • Bases & Subbases
  • Relativization & Separation
  • Connected Sets
• Moore-Smith Convergence
  • Directed Sets & Nets
  • Subsets & Cluster Points
  • Sequences & Subsequences
  • Convergence Classes
• Product & Quotient Spaces
  • Characterizations of Continuity
  • Homeomorphisms
  • Functions to a Product Space
  • Coordinatewise Convergence
  • Open & Closed Maps
  • Upper Semi-continuous Decompositions
• Embedding & Metrization
  • Tychonoff & Urysohn Lemmata
  • Embedding Lemma
  • Tychonoff Spaces
  • Metric & Pseudo-metric Spaces
  • Urysohn Metrization Theorem
  • Locally Finite Covers
  • Characterization of Metrizability
• Compact Spaces
  • Equivalences
  • Compactness & Separation Properties
  • Products of Compact Spaces
  • Locally Compact Spaces
  • Compactification
  • Lebesgue's Covering Lemma
  • Paracompactness
• Uniform Spaces
  • Uniformities & The Uniform Topology
  • Uniform Continuity & Product Uniformities
  • Metrization & the Gage of a Uniformity
  • Cauchy Nets & Extension of Functions
  • Existence & Uniqueness of Completion
  • Uniqueness of Uniformity & Total Boundedness
  • Baire Theorem, Localization of Category, and Uniformly Open Maps
• Function Spaces
  • Pointwise Convergence
  • Compact Open Topology & Joint Continuity
  • Uniform Convergence
  • Compactness & Equicontinuity
  • Even Continuity
• Appendix on Elementary Set Theory