Elementary General Topology. Moore, Theral O. (Theral Orvis). Englewood Cliffs, N. J., Prentice-Hall, 1964, 174 pages.
Mathematical Monthly had this review in its October 1965 issue, page 923:
The author of this carefully written text states in his preface that he has kept in mind especially the undergraduate students who have not yet had advanced calculus. Thus, one of his aims is to develop mathematical maturity. He has been remarkably successful in his part of the task---the rest is up to the teacher and student. The author repeatedly urges the student to try giving his own proof for each theorem before reading the one in the text. Problem sets appear sprinkled through the text, rather than at the ends of chapters. The problem material includes some development of theory; hints and partial solutions to selected problems are given after the problem sets, with an exhortation to the reader not to look at the hints unless it is absolutely necessary. An intelligent, diligent student who complies with the author's suggestions should really develop mathematical maturity.
Explanations and proofs are given in careful detail. Some material on motivation, and the origin of various concepts is included. The author starts with a general topological space (metric spaces appear in Chapter 6) because he feels it is easier for the student to give the first proofs in the general case rather than in a special case where there is much irrelevant information available. Interpretation of results for the case of the real numbers is emphasized throughout. The last three chapters discuss function spaces, nets and convergence, and Peano spaces. The text is suitable for a one-term course or a year course. The reviewer feels that this is a very good text for a first course in topology, and that this course might be at the junior level in one institution and at the graduate level in another.
B. H. Arnold, Oregon State University
Mathematical Reviews had this review in its April 1965 issue, # 4018, page 768:
An undergraduate text with unusual precision, efficiency, and very good problems. The usual material is covered, and two fine chapters on nets and Peano spaces, respectively, are added. All concepts are well-motivated and studied to the extent practicable on this level.
J. Mayer (Albuquerque, New Mexico)
Prentice-Hall the publisher writes:
The author's purpose in preparing this work has been to provide a systematic survey of the standard topics of general topology which the beginner can follow with minimum effort and maximum results.
His several years experience in teaching topology to both graduate and undergraduate students convinced him that: the early theorems are easier for the beginner to prove for general topological spaces than for metric spaces or the space of reals and that the best understanding is achieved most easily when the beginner has a knowledge of abstract algebra to prepare him for the concept of abstract space.
These convictions explain the order in which he presents his material: elementary set theory, topological spaces, metric spaces, product spaces, function spaces, nets, and Peano spaces.
Applications to the space of reals are stressed particularly. Later chapters use simple applications of the Hausdorff Maximal Principle to prove the Tychonoff Theorem and Hahn-Mazurkiewicz Theorem.
Throughout the book, the author enlists the reader's participation, encouraging him to prepare his own proofs of theorems and inviting him, then, to compare them with the proofs provided.
To give the reader almost simultaneous introduction to principles and their applications, the author also integrates problem materials with the text, rather than grouping it at the ends of chapters.
In toto, what sets his treatment apart is the fact that while it is addressed to the reader with a minimum background, it covers a breadth and depth of topics characteristic of more advanced works in the field.
Prentice-Hall, the publisher