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I am a university graduate with a B.S. in mathematics who has been developing software for the past 8 years. I recently discussed a mutual interest in topology with a friend who is just about to complete his degree. Due to time constraints both of us missed our chance to take a topology course during our undergraduate studies, so we have decided to make an independent study of the subject once he finishes school, and are looking for a book to guide us.

Ideally I am looking for a book that...

  • is suitable for our math background
  • is well laid out and not too difficult to follow (i.e. is suited to self study)
  • does not assume prior knowledge
  • is thorough enough to enable future study of topics within the field
  • contains plenty of exercises
  • isn't sparse on diagrams where they are appropriate
  • doesn't waste much time on overly-specific material (I'm the type that likes to prove things about the determinant, not calculate thousands of them)

What topology texts would be appropriate for our self study?

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I had great pleasure working through Munkries, doing all the exercises. I'm not sure what you want in terms of "foundational introduction" but his (extensive) first chapter will be an excellent ramp-up getting back into shape. In particular, before even getting into topology, his exercises leading to the proof that (even without the axiom of choice) there exists an uncountable, well-ordered set were really nice, and should not be skipped.

There are not a lot of diagrams, but there are some where needed.

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    $\begingroup$ +1, but it's Munkres, not Munkries. (And he also has an algebraic topology book, which I guess is not the one you mean.) $\endgroup$ – Hans Lundmark Jun 14 '18 at 20:09
  • $\begingroup$ By "foundational" I mean both that it shouldn't assume prior knowledge of topology, and that I could dig deeper into the field upon completing it. That it is a thorough introduction, in other words. I'll edit my question to that effect. $\endgroup$ – Conduit Jun 14 '18 at 22:43
  • $\begingroup$ He also has a differential topology book . . . but I think it is obvious from the context which book the responder means, and is certainly what will appear in a Google search. If they meant algebraic topology, they would have said so. I would add that chapters 6/7 can probably be skipped from Munkres, to the OP. $\endgroup$ – John Samples Jun 14 '18 at 23:21
  • $\begingroup$ Settled on this, and loving it so far. Thanks! $\endgroup$ – Conduit Jul 16 '18 at 14:59
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I'd recommend Crossley's Essential Topology. It's pitched at a great level, and very easy to read. It covers the basics of topological spaces, homotopy, homotopy groups, simplicial homology, and singular homology.

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Munkres in an excellent choice. You might also want to consider Willard's "General Topology." Two inexpensive options worth mentioning are Gemignani, "Elementary Topology" and Mendelson, "Introduction to Topology." Both are available from Dover press and are probably at a somewhat more basic level than Willard or Munkres.

My first exposure to topology was with Bourbaki, "General Topology." It did not go well. I finally learned from Kelley, "General Toplogy." Some think Kelley is difficult. I would disagree, but I would generally not recommend it as a place to start, especially after a bit of a layoff.

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