TITLE: Undergraduate Topology
QUESTION [17 upvotes]: I am developing an introductory topology course for undergraduates, and I am wondering what topics to cover.  At my institution, real analysis is not a prerequisite for the course, so it is more than likely that the intended audience has not been exposed to this material.  Does anyone have any suggestions?

REPLY [2 votes]: I think it depends on your students' backgrounds and goals. You mention that real analysis is not a prerequisite for the course, but what are the prerequisites? Can you assume any or all of linear algebra, multivariable calculus, discrete mathematics, a transition to proofs course and abstract algebra? Are your students interested in applications of topology outside of mathematics or within mathematics, or would they prefer to learn the subject for its own sake (or all three)?
For instance, you can give the course an analysis flavor by focusing on metric spaces and general topology. On the other hand, you could go straight to homology (see Peter Giblin's Graphs, Surfaces and Homology) if your students have taken / are concurrently taking a course in group theory, or you are prepared to teach the requisite group theory during the course itself.
If you'd like to focus on general topology with applications outside of mathematics, Introduction to Topology: Pure and Applied is a nice option.
Computational topology, as mentioned in another answer, could be an option for students with strong applied interests. Robert Ghrist has an intriguing draft text on applied algebraic topology available at his website, which could be a source of inspiration. There's also a question on teaching undergraduates computational topology at the Mathematics Educators Stack Exchange, although it's aimed at engineering students rather than mathematics majors.