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I want to study an advanced modern book on topology, but I couldn't find any. I've already studied the first chapters of Munkres' book, but it is not as advanced as books such as Engelking's topology, but on the other hand Engelking's topology is not really modern.

Is there any advanced modern topology textbook?

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  • $\begingroup$ You're using backwards apostrophes, which is causing formatting issues. $\endgroup$ – Ben Grossmann Jul 13 '15 at 14:32
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    $\begingroup$ Not modern, but there is still nothing better than Willard. $\endgroup$ – John McGee Jul 13 '15 at 14:33
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    $\begingroup$ What do you mean by "advanced"? Munkres is generally views as a very good introductory text, and has majority (if not nearly all) of the information Engelking's book has, just in a different order. $\endgroup$ – anakhro Jul 13 '15 at 14:41
  • $\begingroup$ By advanced I mean a graduate style book,not an introductory one @anakhronizein $\endgroup$ – Milad Math Jul 13 '15 at 14:49
  • $\begingroup$ In case the other comments and answers aren't what you're looking for, and you really do want hard-core general topology, see my comments at Reference for set theoretic topology and perhaps try to get Brian M. Scott's attention for additional advice. $\endgroup$ – Dave L. Renfro Jul 13 '15 at 18:15
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Munkres' book contains almost all of the point-set topology that a typical mathematician (or even a typical topologist) would ever need to know. If you've finished the first few chapters of Munkres, now is probably a good time to drop point-set topology for a while and start learning algebraic topology, e.g. from Hatcher's Algebraic Topology. You can return to Munkres to learn additional topics in point-set topology as the need arises.

If you really want to be a completest about point-set topology, there are only a few topics that aren't in Munkres that might be worth learning about. Most of these are pretty esoteric, to the point that a working topologist can probably get away without learning about any of them. I don't know of any modern book that covers them all:

  1. Nets and their role in point-set topology. See Section 6 of Bredon's Topology and Geometry for a coherent treatment.

  2. Direct and inverse limits in the category of topological spaces.

  3. Cantor spaces and Brouwer's theorem.

  4. Munkres gives short shrift to proper maps, and it's worth learning what these mean geometrically and what can be said about them.

  5. Ends of a space and the end compactification.

  6. Basics of Polish spaces and the Baire space.

  7. Some basics of continuum theory.

  8. A little bit more dimension theory.

  9. Conceivably something about uniform spaces.

Again, my advice would be to not worry about any of this stuff for right now. Just proceed to algebraic topology.

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  • $\begingroup$ Alternatively, depending on your mathematical tastes and interests, start learning some serious set theory. $\endgroup$ – Brian M. Scott Jul 13 '15 at 19:18
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    $\begingroup$ @BrianM.Scott That's a good point -- basic topology really does lead to a variety of different places, not just algebraic topology (though I think of algebraic topology as the next course in a "topology sequence"). But set theory would be a very reasonable place to go next, as would measure theory (if OP doesn't already know it), functional analysis, differentiable manifolds, or some sort of geometric topology. $\endgroup$ – Jim Belk Jul 13 '15 at 19:36
  • $\begingroup$ Regarding set-theoretic topology (what Brian M. Scott's comment suggests to me), M KhodaE might be interested in the very recent article Memories of Mary Ellen Rudin [AMS Notices, June-July 2015]. For those who might not know, Mary Ellen was the wife of Walter Rudin, the author of the well-known textbooks in analysis (Principles of ..., Real and Complex Analysis, and Functional Analysis). $\endgroup$ – Dave L. Renfro Jul 13 '15 at 20:40
  • $\begingroup$ It helped me a lot, thank you so much @JimBelk and Mr Scott and Mr Renfro $\endgroup$ – Milad Math Jul 23 '15 at 11:49
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Dugundji's Topology is a really fine book, much better than Munkres in my opinion (and Hatcher's opinion as well if you look here http://www.math.cornell.edu/~hatcher/Other/topologybooks.pdf

As many other people, I find Munkres pedestrian and uninspiring.

Anyway, Dugundji is not modern, however most of the core of general topology is quite an established subject hence apart from minor changes of notation you should be fine. It is a comprehensive book and self-contained book. I find his writing very clear and his proofs not too terse or too clever.

While it starts from the basic, it avoids the doughnut-mug motivational blurb and starts with topological spaces in their earnest (rather than e.g. introducing the concepts first for metric spaces).

I am not familiar with the whole book, but from what I have seen so far some possible issues are

  • the volume is out of print so you'd have to find a library/second hand copy
  • exercises have no solutions
  • he does not discuss separation axioms weaker than Hausdorff
  • he does not discuss compact spaces which are not Hausdorff
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  • $\begingroup$ Judging from the index and table of contents most of what Jim mentioned is in the book. However I am not familiar with those topic myself so better check if you are interested in something in particular. $\endgroup$ – GFR Jul 13 '15 at 15:24
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To complete the list of excellent mentioned texts, try topology without tears

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