Learning Topology
Since the discussion has broadened from the original question to include a wider range of topology books, let me add one more. This is an algebraic topology book by Tammo tom Dieck published just a year ago with the canonical title "Algebraic Topology". Its viewpoint is fairly homotopy-theoretic, as in May's book, and it has a similar density coefficient that some commenters here seem to like. What really impressed me about the book is that in the last few chapters the author manages to give the first ever non-spectral-sequence proofs of some deep and fundamental theorems like Serre's theorem that the homotopy groups of spheres are finitely generated, and Serre's calculation of all the non-torsion. Another is the Hirzebruch signature theorem, the very last theorem in the book. These results are 50 years old, yet apparently no one had previously seen how to prove them without spectral sequences. Of course, spectral sequences are important things that serious topologists should know about, and their use cannot always be avoided, but it's illuminating to see when they are needed and when they are not. Whenever I get around to a second edition of my book I'll have to include tom Dieck's new approach, and I think one can go even further and develop the basic framework of rational homotopy theory without spectral sequences.
It's too bad that math books aren't like Google Maps where one can zoom in or out to get the level of detail and density one wants, or switch between satellite and map views to include or omit things like examples and informal discussions of ideas and motivation. Maybe someday this sort of thing will be possible with electronic books.
A self study course I can recommend for topology is Topology by JR Munkres followed by Algebraic Topology by A Hatcher (freely and legally available online, courtesy of the author!). But that is if you want to be able to really do the math in all its glorious detail. Basic Topology by MA Armstrong is a shortcut and a very good one at that.
The closest I can get to what you are asking for here is Network Topology. Is that what you mean? In that case you should be probably be looking at topological graph theory. Wikipedia also tells me that something called Computational Topology exists, but that is probably not what you are looking for.
Hope that helps!
If you meant general topology, I recommend Munkres's Topology.