TITLE: Learning Topology
QUESTION [23 upvotes]: EDIT (Harry):  Since this question in its original form was poorly stated (asked about topology rather than graph theory), but we have a list of Topology books in the answers, I guess you should go ahead and post with regard to that topic, rather than graph theory, which the questioner can ask again in another topic.
EDIT (David): The original question was asking for places to learn topology with an eye towards applying it to computer science (artificial neural networks in particular)

REPLY [2 votes]: This question originally asked about places to learn topology with an eye towards applying it to programming. The OP then clarified that he was interested in Artificial Neural Networks. Most answers have been about which textbooks are good for topology, but none address connections to computer science. One mentions computational topology, which seeks to develop efficient algorithms for solving topological problems, e.g. computing homology and homotopy. This doesn't seem to be what the OP was after, since he wanted to apply topology to computer science and not the other way around.
As an algebraic topology PhD student who's simultaneously getting a masters in CS, I have thought quite a bit about ways to combine the two. I've seen talks in which algorithms are developed to apply homology to detect information about graph properties, but these are mostly useless in practice because they're so slow. I haven't seen any algorithms using more than $H_1$ except in the work of Robert Ghrist, who uses sheaf cohomology and similarly high-brow mathematics to do pretty impressive things with sensor networks and engineering (e.g. flows). He seems to be the primary pioneer in this new field and calls it Applied Topology. He recently updated his website to include lots of powerpoints and pdfs to help a beginner get into this field. Here is the link. I've also seen some conferences about Applied Topology in the works, so keep your eyes open for them if you're interested in this field.