Attribute | Rank |
---|---|

Difficulty | 4 |

Proof Quality | 4 |

Readability | 5 |

Self-Study | 5 |

Good for Teaching? | 4 |

Quality of Exercises | 5 |

- 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