Resources for self-study of general topology
Solution 1:
A great resource for self-study in Topology is James Munkres' Topology. You can "preview" the text and it contents at the link given. It's divided into two sections, the second being algebraic topology.
You'll also get an overview of Topology (Wikipedia), its branches and the topics related to those branches/subfields, to better hone in on your area of interest. There are a number of references cited, some with links, along with helpful resources that may aid in your self-study.
A good follow-up to Munkres, and more in the direction of your interests, I presume, is then (of course) Algebraic Topology by Allen Hatcher (freely and legally available online, courtesy of the author! The link takes you to Hatcher's home page, where you can access his text, as well as supplementary resources, e.g. notes in point-set-topology.) You might want to peruse Hatcher's text to get a better idea what preliminary material you should cover in Munkres; that might cut down on the amount of Munkres you'd want to cover.
Another sound intro to Topology is A First Course in Topology by J. McCleary (again, you can preview the text at the given link).
Solution 2:
I would have to disagree slightly about the Munkres recommendation; It is a nice book, and the second edition was what was used when I studied topology at university. However, it may be a bit hard to use for self-study, since it is concerned very much with going though results and definitions, and does not go much into the motivations of those definitions, or their applications. A good lecturer can fill those in, but unfortunately I did not have one of those, so the book was not very useful to me. The same might be true when it's used for self-study.
I can recommend Ronald Brown's "Topology and Groupoids", however, which in my opinion manages to combine the "abstract" nature of topology with a concrete understanding very well. It is a bit idiosyncratic in that it is based more on category theory than set theory, but if what I've understood about the contemporary development of topology is right, that is the direction that the field is moving in anyway.
I have also heard good things about Bredon's "Topology and Geometry" (from people whose judgment I trust in these matters, that is) but I haven't studied it properly yet myself.
Solution 3:
Another really good resource for topology is "Introduction to Topology and Modern Analysis" by George Simmons. I used it alot in graduate school.
Solution 4:
Maybe have a look at topology without tears I am learning Topology right now using this book and found it quite helpful. It is free and there is an accompanying video series on his website.
Solution 5:
Fred H . Croom, Principles of Topology
See: Croom’s:Principles of Topology
Here are partial list of the chapters:
- Introduction
- Line and the Plane
- Metric Spaces
- Topological Spaces
- Connectednes
- Compactness
Looking at the book,the layout is beautiful and the proofs are clearly written showing you how to do the exercises.
Also another professor on another site suggested it over Conover’s A First Course on Topology which I currently use because Conover’s thesis was not shown in American Geneology of Mathematicians
Another excellent book you can get by hard back is
Bert Mendelson,Introduction to Topology
See: Mendleson’s Introduction to Topology
It is heavy on metric spaces.
Here is a list of the chapters:
1.Theory of Sets
2.Metric Spaces
3.Topological Spaces
4.Connectedness
5.Compactness
His proofs of many of the theorems are easily understood and the problems he gives should not be to hard to do.
Chapter 4 has a section on homotopic paths.Homeomorphism is mentioned in Chapter 3.
This is a basic text in Topology.