## Ranking

All ranking is done on a scale of 1-5, with 5 as the best.
Attribute Rank
Difficulty 4
Proof Quality 4
Self-Study 5
Good for Teaching? 4
Quality of Exercises 5

## 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