John L. Kelley’s *General Topology *is a book I discovered *after* my time at university, while searching for an established reference text for the subject. *General Topology *is regarded as a classic guide to the discipline, and I agree with this consensus. The book features extensive coverage of topology from basics to very advanced theory, and all of it is delivered in a refreshingly readable voice that sacrifices no rigor for it.

With regard to practical use for study, I have a few points. Firstly, almost every theorem in the book is fully proven immediately after statement. Exercises are provided at the conclusion of each chapter, and each of these problems is itself a task of proof, so curious readers should have some familiarity with formal proof methods. These problems vary in length, and some are large enough in scope to constitute small courses in other mathematical theory, as they relate to the topology at hand in the chapter preceding.

Though the book on its own serves wonderfully as a reference, Kelley makes very useful addenda throughout the main body to point out original creators of the concepts covered. These include directions to find the source documents as well as remarks on their historical significance and contribution to the realm of study. Students seeking multiple references and first-hand coverage will find these particularly helpful.

A final note: the book has several published editions. The one pictured here is the D. Van Nostrand Company hardcover version (1955), no longer in print. There also exists a Dover softcover edition which is identical in terms of content, and is much more affordable than the Springer hardcover release.

### Review

## Prerequisites

### set theory and logic, proofs, basic mathematical analysis

## 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

### Attributes

Difficulty | 4 |

Good for teaching? | 4 |

Proof Quality | 4 |

Quality of Exercises | 5 |

Readability | 5 |

Self-Study | 5 |