If you want to learn topology wholesale, I would recommend Munkres' book, "Topology", which goes quite far in terms of introductory material.
However, in terms of what might be useful for physics I would recommend either:
- Nakahara's "Geometry, Topology and Physics"
- Naber's "Topology, Geometry and Gauge Fields: Foundations"
Personally, I haven't read much of Nakahara, but I've heard good things about it, although it may presuppose too many concepts. I've read selections of Naber and it seems fairly well written and understandable and starts from first principles, but again, it may not focus as much on the fundamentals, if that's what you're looking for.
This post imported from StackExchange Physics at 2014-04-05 16:34 (UCT), posted by SE-user Nilay Kumar