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?

7$\begingroup$ Welcome to MO! Since this question does not have one right answer, but rather asks for a list of suggestion or varying opinions it should be ask in Community Wiki mode. To achieve this, please 'edit' the question (button below the text of the question) and tick the appropriate box and save this edit. (I also flagged for moderators to do this in case you do not see the request in time, this has implications for all answers, or should have difficulty doing this.) $\endgroup$ – user9072 Aug 22 '12 at 16:23

6$\begingroup$ This might have been a nice question for the new Mathematics Educators Stackexchange site. $\endgroup$ – J W Apr 23 '14 at 16:57

2$\begingroup$ @JW: I second you, this question is interesting but really belong to Math Educators SE. Is it possible to migrate it? I don't see the option anymore. $\endgroup$ – Benoît Kloeckner Apr 23 '14 at 18:44
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.

2$\begingroup$ Ronald,you don't get enough credit for your very inventive and original attempts to reform a basic course in topology. You should get more. $\endgroup$ – The Mathemagician Nov 18 '12 at 5:07

$\begingroup$ @The Mathemagician Many thanks for the comment! One person has called my exposition "idiosyncratic". I suppose that means I never assume an account of topology, or mathematics, is a kind of sacred text  "paths" have to be of length 1, "topologies" are defined in terms of open sets, "groups" are the fundamental idea, ...An exegesis shows one has to consider how the ideas arose. see pages.bangor.ac.uk/~mas010/einst.html, and also try to make the abstract ideas "concrete". See also my Galway exposition on my preprint page, for "anomalies in algebraic topology". I keep trying! $\endgroup$ – Ronnie Brown Jan 29 '16 at 22:08
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.

2$\begingroup$ I second the suggestion to focus on lowdimensional topology. Some particularly fun parts of lowdimensional topology for a first course are: classification of surfaces (including Part 1 of Conway et al "Symmetry of Things"); some baby knot theory (Reidemeister moves, quandle invariants like the number of threecolorings of the knot diagram, and also some invariants that are not diagrambased). It is also worth telling the students without proof that intuition from low dimensions often fails as you move higher up. Exotic smooth structures and nonsmoothable manifolds come to mind. $\endgroup$ – Theo JohnsonFreyd Aug 22 '12 at 17:57
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.

2$\begingroup$ +1 for "to motivate the pointset definitions, you had better start with metric spaces. (I'm not saying you should do pointset topology, but that if you do it, metric spaces should come first.) $\endgroup$ – Andreas Blass Aug 22 '12 at 22:57
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.
EDIT: The homepage link doesn't work any more. The closest thing to it that we make publically available seems to be this (which may only last for the current year).
More edit: Here's the weekbyweek lecture schedule (2 hours of lectures a week):
Topological spaces
Surfaces
Surfaces and Surgery
Characterising Surfaces
Graphs on Surfaces
Graphs and Map Colouring
Graphs and Map Colouring
Knots and Links
The Alexander Number of a Knot
The Alexander Group of a Knot
The Alexander Module
The Alexander Polynomial
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.

$\begingroup$ Any particular reason for not going into knot theory? $\endgroup$ – J W Apr 24 '14 at 19:08

2$\begingroup$ @JW: I like knot theory, even worked on it as an undergrad. However, knot theory has a few problems that make it unsuitable for the first course in topology for people with limited math background. First, application to other branches require too much depth into the subject; not presenting applications would make knot theory look artificial (compare with immediate apps of degree of maps or winding numbers.) Second, meaningful exposure would require lengthy side tours, such as explaining what groups are. Third, not easy to generalize into higher dimensions, even from $S^1$ to $S^2$ embeddings. $\endgroup$ – Michael Apr 24 '14 at 19:54
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.
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.