a good modern topology book 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?
 A: 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:


*

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

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

*Cantor spaces and Brouwer's theorem.

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

*Ends of a space and the end compactification.

*Basics of Polish spaces and the Baire space.

*Some basics of continuum theory.

*A little bit more dimension theory.

*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.
A: 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 

A: To complete the list of excellent mentioned texts, try topology without tears
