My first encounter with a topology textbook was not a happy one. I was in grad school knowing I’d leave in a few months to start med school and with plenty of time on my hands and enough money to do what I wanted. I’d always liked math and had taken calculus, including advanced and differential equations in college. Grad school and quantum mechanics meant more differential equations, series solutions of same, matrices, eigenvectors and eigenvalues, etc. etc. I liked the stuff. So I’d heard topology was cool — Mobius strips, Klein bottles, wormholes (from John Wheeler) etc. etc.
So I opened a topology book to find on page 1
A topology is a set with certain selected subsets called open sets satisfying two conditions
l. The union of any number of open sets is an open set
2. The intersection of a finite number of open sets is an open set
In an effort to help, on page two the book provided another definition
A topology is a set with certain selected subsets called closed sets satisfying two conditions
l. The union of a finite number number of closed sets is a closed set
2. The intersection of any number of closed sets is a closed set
Ghastly. No motivation. No idea where the definitions came from or how they could be applied.
Which brings me to ‘An Introduction to Algebraic Topology” by Andrew H. Wallace. I recommend it highly, even though algebraic topology is just a branch of topology and fairly specialized at that.
Because in a wonderful, leisurely and discursive fashion, he starts out with the intuitive concept of nearness, applying it to to classic analytic geometry of the plane. He then moves on to continuous functions from one plane to another explaining why they must preserve nearness. Then he abstracts what nearness must mean in terms of the classic pythagorean distance function. Topological spaces are first defined in terms of nearness and neighborhoods, and only after 18 pages does he define open sets in terms of neighborhoods. It’s a wonderful exposition, explaining why open sets must have the properties they have. He doesn’t even get to algebraic topology until p. 62, explaining point set topological notions such as connectedness, compactness, homeomorphisms etc. etc. along the way.
This is a recommendation not a review because, I’ve not read the whole thing. But it’s a great explanation for why the definitions in topology must be the way they are.
It won’t set you back much — I paid. $12.95 for the Dover edition (not sure when).