A serious and unusually well-disciplined graduate text: broad in scope, mathematically exacting, and refreshingly resistant to the usual hand-waving that too often mars topology expositions.
I found Richard Murdoch Montgomery’s treatise to be an admirably serious piece of mathematical writing. It is not merely comprehensive in title; it earns that description by moving with real competence from the basic language of point-set topology to the more demanding terrain where general topology meets algebraic and geometric questions. The book’s chief virtue is its discipline: definitions are stated with precision, hypotheses are not silently weakened, and the proofs proceed in a manner that assumes intelligence without indulging in obscurity.
What impressed me most was the architectural sense of the exposition. The foundational chapters are not treated as an apology for the later material, but as a proper logical base: separation axioms, compactness, connectedness, quotient constructions, and function-space ideas are handled with the kind of care that makes subsequent theorems genuinely feel earned. In a field where authors too often leap too quickly from intuition to assertion, Montgomery is commendably exact. I was especially pleased by the way the text keeps returning to the interplay between local and global structure, so that abstract notions are not left hanging as isolated definitions.
The “frontiers” portion of the book is, happily, not decorative branding. The later chapters show ambition without sacrificing rigour, and the author manages the graduate-level balance between breadth and depth better than most survey texts. I appreciated the clean internal cross-referencing and the sense that each result has been checked against the surrounding theory rather than inserted for effect. If I have a mild reservation, it is only that the breadth of the book occasionally leaves me wishing for a few more worked examples in the most abstract passages; yet this is a small and almost affectionate complaint in a work so evidently committed to mathematical honesty.
As a graduate exposition, this is precisely the sort of book I trust: serious, structurally coherent, and mathematically literate at every stage. It will not flatter the reader, but it will reward the reader who is willing to think carefully. That, in topology, is the highest compliment I can offer.
Who should read this
I should recommend this to graduate students and researchers who want a rigorous, wide-ranging topology text that respects proof and definition equally. It will suit readers who already have some maturity in analysis or abstract algebra and want a dependable guide beyond the elementary course.
A personal note from Kowalski
I closed it with the rare satisfaction of having read a book that knows exactly what topology is, and what a topology text ought to do.
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