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?

In the 1970s I developed an undergraduate course on knots, (source book was by Crowell and Fox) to replace general topology and homology, as it was very easy for students to understand the point of the course, there were interesting relations with group theory, and lots of specific calculations and other things to do. The course was eventually taken over by others, and resulted in a book, Knots and Surfaces, by N.D. Gilbert and T. Porter which had good reviews. For me, it led to giving popular talks on "How mathematics gets into Knots", and eventually to the exhibition you can see on the web site for the Centre for the Popularisation of Mathematics. In these talks I could also talk about mathematics, including, for example, the importance of analogy in mathematics. This led to one boy at a talk for children, some aged 12, asking: "Are there infinitely many prime knots? " Wow! So giving this undergraduate course has led to all sorts of fun and rewarding things! My copper pentoil knot used with string to demonstrate the ideas of the fundamental group has also travelled to many countries, see for example the pdf of a William J. Spencer Lecture in Kansas, April, 2012. 


I've found that doing lowdimensional manifold topology is very appealing to undergraduates. I used the "Topology Now!" text by Messer and Straffin and, while the text isn't perfect, the approach was wonderfully successful. 


I second Aeryk's suggestion to focus on lowdimensional topology. More generally, I think that algebraic topology can be more exciting than pointset topology. That said, when I was an undergraduate, I do remember being quite excited about the Bourbaki program and pointset definitions and so on. I remember one "first course in topology" that alternated days: lowdimensional topology on even days and pointset on odd. Except the pointset portion began with set theory, cardinal and ordinal numbers, and the axiom of choice; then moved on to metric spaces; and only then introduced pointset topology. I basically think that to motivate the pointset definitions, you had better start with metric spaces. If on the other hand you focus more on broadlydefined algebraic topology, then in addition to the lowdimensional topology of manifolds (surfaces, knots, etc.), another good topic is Brower fixedpoint theorem as an application of fundamental group functor on pointed spaces. Perhaps, if you are very ambitious, you can prove that 2dTQFT = commmutative Frobenius algebra, and talk more generally about cobordism equivalence. Oh, and especially given the recent sad news, be sure to include a little Morse theory and Outside In. 


If the intent is to provide breadth, then many of the suggestions others have made are quite appealing, especially if it is made clear what branches of topology are being introduced and what a student should do outside of class to develop depth in any or all of the branches. If the intent is to provide depth, there are likely several texts out there, one for each branch, with suggestions. I remember covering Munkres first course in Topology starting with chapter 2; even though we skipped over the set theory and foundations, I was intrigued enough by them to study set theory and foundations while in graduate school. Although the class did not go all the way through the book that first semester, we got exposed to quite a bit, and I developed more of a taste for formalism from that class more than from any other that I took as an undergraduate. If the intent is to provide both depth and breadth, I suggest part of it be run as a student seminar. A later topology course I took had me present Sard's theorem; if nothing else came from that course I at least know how to prepare to explain Sard's theorem for my next opportunity. Gerhard "And This Was Decades Ago" Paseman, 2012.08.22 


I think you could do a lot worse than to focus on modern applications by using the texts of Edelsbrunner and Harer and/or Zomorodian on computational topology as touchstones and an avenue towards current work in topological data analysis. These books are selfcontained treatments that focus on Morse theory and homology over $\mathbb{Z}/2\mathbb{Z}$, and there are a lot of materials and software available for a course to be built from. 


As a resource for lowdimensional topology, I would suggest The shape of space by Jeffrey Weeks. It covers how to build compact twomanifolds and some three manifolds. Exercises include playing tictactoe on surfaces and forming sums of surfaces. It then moves on to some three dimensional topology with a heavy focus on attempting to visualize the spaces. 


We give a Geometry and Topology course at Macquarie for students with no real analysis, using notes written by a colleague. Here is the homepage for the course, so you can get an idea of what we do. 


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. 


I would start with degree of maps and show applications to other fields, such as proving that $\mathbb C$ is algebraically closed via polynomial mapping of circles of increasing radii, contracting them onto the unit circle, and computing the degree of the composition map. Then discuss homotopy and prove hairy sphere theorem. Then classification of 2surfaces and Euler characteristics. Perhaps rudimentary Morse theory, starting with classification of surfaces and explaining why $e(S^n)$ depends on parity of $n$. Discussing fundamental groups would be a good idea if and only if the audience have learnt group theory before; otherwise you'd risk to go astray from visual into algebra. Keeping things visual helps; I wouldn't go into knot theory though. 


Since Real Analysis is not a prerequisite for Topology in your institution, I suggest including a chapter on "Metric Spaces" in the course. Then moving on to a chapter on "Topological spaces", "Product Topology", "Compactness", "Countability and "Separation Axioms (with Urysohn’s Lemma)" and "Connectedness". Advanced topics like The StoneČech Compactification, Urysohn's theorem, Ascoli's theorem, etc can be omitted. This should be enough for one semester, which will give students a fair idea of General Topology. Interested students can learn advanced topics like manifolds, algebraic topology, homotopy etc. 

