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Topology for Analysis

Albert Wilansky
Dover Publications
Publication Date: 
Number of Pages: 
[Reviewed by
Allen Stenger
, on

This is a comprehensive treatise of the portions of topology that are needed for analysis. It covers not just the topology of the real line (which is where we usually first meet topology) but all areas of analysis, including topological groups, function spaces, and functional analysis. The present volume is a 2008 reprint of the 1970 work published by Ginn and Company.

The book is well-done, with clear writing, lots of exercises, and a clean layout with lots of white space and section markings. The book has a unique feature I've never seen before: a 40-page appendix that tabulates theorems and counterexamples. The tabulation is extremely compact and consists of a table for each topological property, telling which other properties this property implies and which it does not imply, with a reference in the literature for each entry. I found it a bit overwhelming and it seems to be much more advanced that the rest of the book, but for people who need this kind of information it could be invaluable. (This table is a very different thing from the Steen & Seebach book Counterexamples in Topology, that gives the actual counterexamples and tells you what they are counterexamples to.)

This book has in mind the same audience as Kelley's General Topology (Kelley wanted to call his book "What Every Young Analyst Should Know"). Wilansky is slightly more abstract and more topological in his approach, while Kelley slants more to the specifics of analysis. Wilansky is more chatty and gives more of the background and motivations of each topic, while in Kelley topics tend to appear in isolation. Wilansky takes a peculiar "handwaving" approach to set theory, where he doesn't really explain anything but claims that it can be explained; this is the weakest aspect of his book. Kelley goes to the other extreme, not only leading off with a Chapter 0 that is mostly about set theory, but including an Appendix which is the first publication of the (then new) Morse-Kelley set theory. Both books devote a large amount of space to exercises; Wilansky's tend to be short items, which Kelley's are longer and might be better called "projects".

Both books are excellent texts in the subject. Other things being equal, Wilansky has a slight edge because it has a cleaner layout, is 15 years newer, and is now available in an inexpensive Dover edition.

Allen Stenger is a math hobbyist, library propagandist, and retired computer programmer. He volunteers in his spare time at, a math help site that fosters inquiry learning. His mathematical interests are number theory and classical analysis.

1 Introduction
1.1 Explanatory Notes
1.2 n-Space
1.3 Abstraction

2 Topological Space
2.1 Topological space
2.2 Semimetric and metric space
2.3 Semimetric and metric topologies
2.4 Natural topologies and metrics
2.5 Notation and terminology
2.6 Base and subbase

3 Convergence
3.1 Sequences
3.2 Filters
3.3 Partially ordered sets
3.4 Nets
3.5 Arithmetic of nets

4 Separation Axioms
4.1 Separation by open sets
4.2 Continuity
4.3 Separation by continuous functions

5 Topological Concepts
5.1 Topological properties
5.2 Connectedness
5.3 Separability
5.4 Compactness

6 Sup, Weak, Product and Quotient Topologies
6.1 Introduction
6.2 Sup topologies
6.3 Weak topologies
6.4 Products
6.5 Quotients
6.6 Continuity
6.7 Separation

7 Compactness
7.1 Countable and sequential compactness
7.2 Compactness in semimetric space
7.3 Ultrafilters
7.4 Products

8 Compactification
8.1 The one-point compactification
8.2 Embeddings
8.3 The Stone-Cech compactification
8.4 Compactifications
8.5 C- and C*-embedding
8.6 Realcompact spaces

9 Complete Semimetric Space
9.1 Completeness
9.2 Completion
9.3 Baire category

10 Metrization
10.1 Separable spaces
10.2 Local finiteness
10.3 Metrization

11 Uniformity
11.1 Uniform space
Il.2 Uniform continuity
11.3 Uniform concepts
11.4 Uniformization
11.5 Metrization and completion

12 Topological Groups
12.1 Group topologies
12.2 Group concepts
12.3 Quotients
12.4 Topological vector spaces

13 Function Spaces
13.1 The compact open topology
13.2 Topologies of uniform convergence
13.3 Equicontinuity
13.4 Weak compactness

14 Miscellaneous Topics
14.1 Extremally disconnected spaces
14.2 The Gleason map
14.3 Categorical algebra
14.4 Paracompact spaces
14.5 Ordinal spaces
14.6 The Tychonoff plank
14.7 Completely regular and normal spaces

Appendix: Tables of Theorems and Counterexamples