I agree with the reviewer before me: this book sets the bar for poor mathematical exposition. Its passages are little more than shoddily constructed rambles.



The tragedy is that Lawson does have some interesting ideas about how to approach topology. His geometric emphasis can indeed provide useful intuition, while "projects" contained in the supplementary exercises are engaging and provide useful experience.



But the good pedagogical concepts underlying his work do not excuse the terrible writing that fills it. In fact, no word better characterizes Lawson's Topology than "lazy". As the previous reviewer mentioned, Lawson has an almost pathological habit: he writes pages of non-rigorous and poorly justified arguments, then literally "leaves the details as an exercise." These "details" are frequently tedious computations unrelated to topology. Meanwhile, the "proofs" often aren't proofs at all; they nebulously cite previous arguments ("we can adapt the argument of section X.XX") in a way that mires the reader in page-turns and confusion. Such garbage has made my classmates -- some of the best students at a top-10 university known for its undergraduate math -- uniformly revile this text.



I'll make my advice simple: if you are a self-learner, do not buy this book. If you are a professor, do not use it. If you are Terry Lawson, halt the publication of this mess until you can make it useful.