I'm assuming that the students are not familiar with point-set topology and it's the first course in topology for them. I'd recommend a combination of Munkres and Intuitive topology by V. V. Prasolov . There will be a great deal of precision and intuition all together.
I'd recommend a combination. Topology by Munkres for the point set stuff, and Algebraic Topology by Hatcher for the algebraic topology. You get all the advantages of two more specialized textbooks, and since Hatcher's text is free, your students won't need to buy two textbooks.
I am an undergraduate student. I think that when you begin to study a new subject it is better to start from books not too broad. For a basic course in topology, I recommend these books (based on my experience as student) J. Dugundji, Topology; C. Kosniowski, A first course in algebraic topology; L.C. Kinsey, Topology of surfaces .
I'm fond of Wilson Sutherland's book Introduction to Metric and Topological Spaces . It covers topics such as completeness and compactness extremely well. In particular, the motivation of compactness is the best I've seen. (It doesn't do any algebraic topology, though.) I just taught a class using it, and it was generally well liked.
From several points of view i.e. group theory and computability and visualization I suggest 3 books: 1.Topology and Groupoids Prof Ronnie Brown Chapter 1-4 are one of the best approaches to the topology I have ever seen. The students learn the concepts fast, their theoretical language to explicate honed, and their visualization skills improved. From chapter 5 and on it provides one of the most mo…
The notes from when I learned topology were eventually published as a UTX book called "A taste of topology" by Volker Runde. It starts with metric spaces but ends at the same place your intended course.
I am bound to recommend my book Topology and Groupoids, (2006) Ronald Brown, available from amazon.com . An e-version is also available from www.kagi.com for £5. See my web page http://www.bangor.ac.uk/r.brown/topgpds.html for links to reviews. It takes a geometric approach, and at the same time a categorical view, that is, there is an emphasis on constructing continuous functions. The approach t…
Willard's General Topology is my favourite book on point-set topology (together with Bourbaki, but the latter is not suited as course text for several reasons). It also defines the fundamental group, but doesn't really do anything with it. More geometric is Lee's Introduction to Topological Manifolds , it is also very student friendly.
Introduction to Topology: Pure and Applied , by Colin Adams and Robert Franzosa.
Immediately after proving that there is no retraction from the disk onto its circle boundary,
they use degree theory to analyze sudden cardiac death.
There is a chapter on knots, a chapter on dynamical systems, sections on Nash equilibrium
and digital topology, a chapter on cosmology.
A fairly streamlined book, although initially gentle, is Essential Topology by Crossley. It goes up to homotopy and homology. See also Celebrating Swansea University Authors to view Crossley talking about his book.
A book that I find very readable is "Topology" by John G. Hocking and Gail S. Young. I have little teaching experience, but I remember being a student and based on that I believe that a few years ago I would have also liked this book.
I will have to teach a topology course:
it starts in point set topology and ends at fundamental group of $S^1$. In the past I have used two different books: Elementary Topology. Textbook in Problems, by O.Ya.Viro, O.A.Ivanov, V.M.Kharlamov and N.Y.Netsvetaev. A First Course in Algebraic Topology by Czes Kosniowski I like both of these books and my students hate both of them.
So I am thinking, may…
