Topology learning recommendations

Basic ideas of point set topology are an important part of undergraduate mathematics education. The most important ideas are covered in an undergraduate analysis course or course sequence. However, a point set topology course typically deals with the ideas at a somewhat higher level of abstraction than the typical analysis course.

For most people, it is recommended that the study of point set topology be deferred until after you have taken an undergraduate analysis course sequence (or done equivalent self-study), because many ideas of topology generalize concepts seen in the analysis sequence. However, if you have a strong background in theoretical calculus and have a strong background with proofs in general (perhaps acquired in the context of contest math) then it may be more efficient for you to self-study point set topology first and then go over the most important theorems in analysis.

Recommendations

• Topology (2nd Edition) by James Munkres (Amazon link): This is the most standard and likely the best introduction to point set topology. It goes into adequate detail and has clear proofs
• Introduction to Topology (Second Edition) by Theodore W. Gamelin and Robert Everist Greene (Amazon link) is a cheaper book that covers all the important topics, and may not differ that much from Munkres for the most important subtopics.
• Lecture Notes on Elementary Topology and Geometry by I. M. Singer and J. A. Thorpe (Amazon link) includes a very compact introduction to point set topology in its first two chapters. This is not good as a primary learning source, but if you intend to study higher mathematics, it could be a useful book to read from and be inspired by on the side as you work through either of the above books.