Topology is an extension of geometry and built on set theory, it's about continuity. There is a joke that topologists are those who cannot distinguish between a doughnut and a coffee or tea cup. Wikipedia has a [.gif] to demonstrate this. You may have studied open sets, open neighbourhood, interior, closure, boundary, basis, continuity, compactness and connectedness etc. in the analysis course, move on to homemorphism, homotopy and fundamental group.
As in abstract algebra and analysis, several selected books or notes cover more than enough. You may leave a few sections to study at the next stage.
- Armstrong M.A. Basic Topology - Another nice introductory text by Armstrong, he succeeded in developing reader's geometrical insight.
- Crossley M.D. Essential Topology - Undergraduate introductory level, detailed explanation.
- Davis S.W. Topology - It's like lecture notes, many proofs are left as exercises and many exercises are proofs.
- Dugundji J. Topology - Another topology text with quality.
- Engelking R. General Topology
- Gemignani M.C. Elementary Topology
- Henle M. A Combinatorial Introduction to Topology - Plenty of illustrations. Indeed many concepts in this subject can and should be visualised.
- Ikenaga B. Notes on Topology (FREE!)
- McCluskey A. and McMaster B. Topology Course Lecture Notes (FREE!)
- Mendelson B. Introduction to Topology - Very nice introduction to the metric spaces, topological spaces, connectedness and compactness. Recommended for first reading.
- Moller J.M. General Topology [.pdf] (FREE!)
- Morris S.A. Topology Without Tears (FREE!)
- Munkres J.R. Topology - Another nice introduction. Go further than Mendelson's one. Also cover topics like the Tychonoff theorem, metrization theorems, complete metric spaces and function spaces, the fundamental group and covering spaces.
- Singer I.M. and Thorpe J.A. Lecture Notes on Elementary Topology and Geometry
- Steen L.A. and Seebach J.A. Counterexamples in Topology
- Strickland N. Topology Notes (FREE!)
- Thurston W.P. The Geometry and Topology of Three-Manifolds (FREE!)
- Viro O.Y., Ivanov O.A., Netsvetaev N.Y. and Kharlamov V.M. Elementary Topology: Problem Textbook (FREE!) - Over 400 pages of free material.
- Ward T.B. Topology Lecture Notes [.pdf] (FREE!)
- Wilkins D.R. Topology (FREE!)