Prerequisites for Bredon's "Topology and Geometry"? My background in topology is the first 6 chapters of Munkres's "topology" and in algebra Herstein's "Topics in Algebra". Both of them I self studied.
A look at the table of contents of Bredon's "Topology and Geometry" got me to really want to read it (I really like the emphasis on the differential side of things). I don't have any background in differential topology/geometry though...
What are the prerequisites I should be familiar with before tackling this book?  
 A: You should read Milnor's topology from a differentiable viewpoint (two or three times) first, then Bott/Tu. Then you are good to go.
A: In my opinion, if you want to study Bredon's book by yourself, you need a good background in elementary topology and some knowledge of differential geometry and algebraic topology. 
Since you already studied the first chapters of Munkre's book Topology, you shouldn't have any problems studying Bredon's first chapter "General Topology". For chapter 2, Tu's book Introduction to Manifolds may be very helpful, or any other elementary book with exercises and solutions. From there, Bredon's book focuses more on algebraic topology with a geometric flavor. Since you have no previous knowledge of algebraic topology, I would say, spend some time (maybe a couple of weeks) reading books such as Armstrong, Basic Topology, Janich, Topology, or the second part of Munkres'; then study Bredon's book from chapter 3. Alternatively, you can concentrate on Bredon's book and take a look at these three books if you find some passages difficult to understand. 
Finally, it is worth saying, Bredon's book is a challenging one. In my humble opinion, it will take you years to really grasp everything it contains. Good luck.
