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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topology text was Munkres' "Topology," but I definitely wouldn't recommend that one; it's extraordinarily dry, outdated (e.g., in its treatment of paracompactness and metrization theorems), and has a desultory chapter on algebraic topology that just winds up being confusing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?

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    $\begingroup$ I really like "Topology without tears" topologywithouttears.net/topbook.pdf $\endgroup$
    – Kandinskij
    Mar 24, 2022 at 17:17
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    $\begingroup$ @FrancescoScavella: Thanks, I'll take a look. $\endgroup$
    – anomaly
    Mar 24, 2022 at 17:18
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    $\begingroup$ I like Armstrong. $\endgroup$ Mar 24, 2022 at 17:19
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    $\begingroup$ Munkres is good $\endgroup$
    – MPW
    Mar 24, 2022 at 18:02
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    $\begingroup$ Some general suggestions along with some possible texts are given in this answer. $\endgroup$ Mar 25, 2022 at 14:23

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  1. Topology by Munkres.

  2. Introduction to Topology and Modern Analysis by George F. Simmons.

  3. Topology Through Inquiry by Francis Su and Michael SStarbird.

  4. Elementary Topology by Oleg Viro

  5. Counterexamples in Topology by J. Arthur Seebach Jr. and Lynn Steen

http://www-groups.mcs.st-andrews.ac.uk/~john/MT4522/index.html

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    $\begingroup$ Thanks for the answer, but I specifically mentioned Munkres and Counterexamples in Topology as books I wanted to stay away from in the OP. $\endgroup$
    – anomaly
    Mar 25, 2022 at 14:33

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