I am looking for a book on Topology which isn't solely focused on cramming theorems and proofs down the reader's throat: I have already read a significant chunk of Munkres. What I am looking for now is a textbook which can truly cement intuition behind all the basic concepts of the course. If anybody has any recommendations, I would appreciate it. Thank you! (And Happy Holidays!)

(For further context, I will be taking a graduate-level course on Algebraic Topology during Spring. My background in Topology is confined almost entirely to the readings of Munkres over the past week, alongside a brief discussion relating measurable and topological spaces in Real Analysis.)

To provide an example as to what exactly I'm looking for: Paul Halmos' Naive Set Theory. This textbook served a strong role in motivating and therefore teaching the basics to all things Set Theory.

Another example: Folland's Real Analysis: Modern Techniques and Their Applications in comparison with Rudin's Real and Complex Analysis. While I do appreciate the brevity and to-the-point nature of Folland, it can be extraordinarily confusing to a first-time learner: The motivation is lacking, and definitions are sometimes left to the reader to assume understood. Rudin's textbook provides a far more motivated approach to the material, using a direct connection between the study of continuous (and later, Riemann integrable) functions to measurable (and later, Lebesgue integrable) functions. The theory suddenly becomes far more intuitive.

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    $\begingroup$ If you know real and complex analysis then you can find, in that subject itself, much of the intuition of general topology: open and closed subsets and their properties; limit points of subsets; interior points of subsets; closures of subsets; interiors of subsets; the definition of continuous functions; ... When reading Munkres, match those topics as they are introduced back with the versions you learned in analysis. $\endgroup$
    – Lee Mosher
    Dec 23, 2023 at 15:16
  • $\begingroup$ But perhaps this answers your question? math.stackexchange.com/questions/4411944/… $\endgroup$
    – Lee Mosher
    Dec 23, 2023 at 15:22
  • $\begingroup$ Jänich provides lots of intuition. $\endgroup$ Dec 23, 2023 at 15:22
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    $\begingroup$ I was thinking about your question again and it occurred to me that instead of what you asked for, it might be better to focus on the topic of quotient spaces -- read and work through its treatment in several texts. Restricting yourself to quotient space sections of topology texts is a more realistic task for the next few weeks, and this topic will be especially relevant for your algebraic topology course -- see Hatcher Algebraic Topology: I have all the prereqs, so why is this book unreadable for me?. $\endgroup$ Dec 25, 2023 at 7:56
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    $\begingroup$ While looking for something else, I came across Notes on Introductory Point-Set Topology (also at archive.org) by Allen Hatcher (author of your upcoming algebraic topology text). Description of the notes -- Hatcher says they form the first part of a topology course he taught in 2005. $\endgroup$ Dec 28, 2023 at 18:08


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