Someone asked me recently about an introductory textbook on point-set topology, and it occurred to me (despite having a background in topology myself) that I didn't actually have one to recommend. My undergrad topology text was Munkres' "Topology," but I definitely wouldn't recommend that one; it's extraordinarily dry, outdated (e.g., in its treatment of paracompactness and metrization theorems), and has a desultory chapter on algebraic topology that just winds up being confusing rather than useful. Ideally, I'd like to find something along the lines of Hatcher but for a more introductory audience. More specifically, I like that book's general writing style, its preserving the geometric flavor of the subject, and its offering many side topics in optional sections at the end of each chapter.
The whole point of point-set topology at this first level is to introduce various conditions on topological spaces and to explore what consequences they force, mostly through first-principles. It's admittedly hard to make that interesting, especially since I'd also like to avoid a textbook that's mostly a gallery of pathological spaces. (I would happily recommend "Counterexamples in Topology" in general, but not for this specific goal.) On the other hand, I would consider books like Rudin to emphasize the analysis side of things too much for these purposes. I don't want to limit the text to metric spaces, for example.
The correct answer may be to recommend a book like Hatcher anyway, with the goal of picking up the needed basic topology (e.g., general definitions, compactness, connectedness, and metric spaces) from the text as needed. Is there a good alternative?
Best Answer
I really enjoyed https://topology.mitpress.mit.edu "Topology:A Categorical Approach" by Tai-Danae Bradley, Tyler Bryson, and John Terilla
Although the book introduced basic notions of category theory in the first chapter, it's a gentle introduction and leads to basic notions in point-set topology. It has a perk of simplifying some constructions (product/quotient topology) to basic categorical facts. At the end it has a nice little chapter on very basic homotopy theory. The book is short (only 150 page) and gives you just enough point-set topology+category theory to read Hatcher next. One possible drawback is it doesn't emphasize on the analytic aspects (no Urysohn's lemma).