What is topology? Break comes to a close, and you, a renowned mathematics professor, step into a grand lecture hall to deliver the first lecture of the semester on topology. This is an introductory course. Half of the students cannot even pronounce homeomorphism. As you look around the room, a bead of sweat works its way across your brow. All you can think of is the possibility that the entire class will fail, and you will be mocked by the other professors. Then you take a sip of water and pull yourself together. You pick up a fresh (but not too fresh) piece of chalk, write your name across the board--effectively marking your territory--and address the class.

How do you introduce a class of undergraduate students to the field of topology?

I am looking for a creative, but precise explanation of the field and the most fundamental topological concepts. Diagrams and metaphors are welcome.
 A: The way I've always explained topology to my non-math friends is to imagine a piece of putty.  We want to ask ourselves "what stays the same about the putty if we're able to stretch it, bend it, twist it, etc... provided that we cannot glue it to itself or cut it anywhere?"  Another way I like to approach the problem to non-math people is to have them imagine their favorite geometric shape (I'm guessing you want to start from an intuitive standpoint with your students), and I tell them to consider properties we might be interested in: how many pieces does the shape have, does it have holes, how big is it, are there angles, what are their measurements, etc.  I then ask them to consider what are the properties of the shape if we get rid of the notions of size, distance, and angles.  From an intuitive standpoint we might say that we're simply left with holes and connected pieces (i.e. components), but in essence we can infer that Topology is the study of the properties of some geometric object or space that remain the same under continuous deformation.  
This might be one way to introduce the subject, but I suppose it depends on whether this is topology course for majors or non-majors.  If it's a class for non-majors then we can illustrate some of the more advanced ideas in topology with interesting examples: fundamental groups, homology groups, homotopies (even just doing these things with the sphere and torus, and illustrating how they can be used as classification tools might be sufficient).  If the course is for majors then I assume that you'll need to go the point-set theoretic approach with defining a topology on a set as a collection of subsets with global union and intersection properties.  Starting from point-set topology it's important to constantly illustrate both the simple counter-intuitive examples (i.e. non-trivial topologies on finite sets, say), with the more intuitive topologies of spaces we can visualize (manifolds in 1 or 2 dimensions, products of spaces, disjoint unions of spaces, etc.).
A decent book for you to check out (and might be especially appealing since it's freely available online) is Topology Without Tears.  This is a concise book that takes a point-set approach (nothing terribly advanced though, definitely no homology in this text) but makes it quite easy.  Can't beat the price.
A: A professor told me that Topology is the study of continuous functions. I think this is the best explanation for day 1 of a class in topology. We've all heard the coffee-cup vs donut analogies, but these are pretty far afield from a first semester in topology.
In fact, much of the first semester will be spent dealing with (relatively) pathological spaces. Students are familiar with the real numbers, which are more or less well behaved, but the intuition they have here won't apply to non-Hausdorff spaces, for example. 
The first semester of topology will teach students precision in a way they didn't know before. It will expose them to various ways to generalize the spaces they are familiar with, and how these generalizations are useful or interesting. It may be the first time students see (relatively) pathological behavior - such as the trivial topology on the real line where every sequence converges to every number. It will also give them perspective on the history of mathematics - consider the difference between limit-point compactness and compactness. Historically, limit-point compactness came first since it works in familiar spaces. Compactness as we formulate it now came later.
Anyways, I know this is a lot of small things instead of one big interesting thing, but if I were to teach topology at some point in the future, these are the things I would tell students about. I might also mention higher topological concepts like knot theory, coffee-cup vs donut, etc, but I would tell my students that is what they could do in future courses, since it is unlikely these things will be considered in a first semester.
A: I think that topology is born mainly from the need to have a vision of low level of particular mathematical structure, ie a topological space can be understood as a space more low-level than finite-dimensional Euclidean spaces, normed spaces, Hilbert spaces. Many of these spaces, and among the most studied, are clearly all topological spaces. A differentiable manifold is a topological space.
Evidently the properties that define a topology are good and valid for a large number of other spaces.
Historically the infinitesimal calculus concepts are born before, from well known general topology results we can deduce the definition of continuity of a real variable function or more general continuity between metric spaces that is studied in a calculus course, there may be many other examples.
They are also very useful axioms of separation, for example, if a topological space is Hausdorff's, then is true the uniqueness theorem of the limit. There are mixed structures such as topological vector spaces, which are always topological spaces and they are important in functional analysis. Topological spaces are very important in differential geometry. There is a discipline called algebraic topology (but I do not know much).
However, this seems just that the structure of topological space is something small compared with mathematical structures with more properties, which are always topological spaces.
A: Given your description of the class, I assume you are talking about an introductory course in point-set (aka general) topology. I would start by saying something along the following lines: 
Topology was born in response to needs of diverse branches of mathematics: First combinatorial geometry, then differential geometry, complex analysis, real analysis, differential equations (more or less in this order), etc. In modern mathematics, topology is ubiquitous. In fact, one has to work hard in order to identify an area of mathematics where topology is not used. Beyond mathematics, topology makes appearance in natural sciences, such as physics, chemistry, biology, as well as in some social sciences such as economics. In order to serve needs of all these diverse areas topology has to be very general and, hence, rather abstract. Thus, our course will start with very general and abstract definitions for which, at first, you will have  very little to no intuition. Gradually, you will learn how to work with these, but you have to be very patient. We will use examples from the courses in real analysis in order to illustrate the abstract notions which appear in the course. For instance, several topological concepts will be central for our course:


*

*Continuity of functions/mappings. 

*Connectivity. 

*Compactness. 
All three concepts are far-reaching generalizations of definitions and phenomena that you know from real analysis: Continuous functions of one or several variables, the intermediate value theorem, the Bolzano-Weierstrass theorem. In our course we will see how and to which extent these extend beyond the realm of the real line and of $R^n$. 
Another important concept that we will be covering is the product topology. We will see how this generalizes the notion of pointwise convergence of sequences of functions that you discussed in your real analysis class. 
Towards the end of the course we will discuss the quotient topology. This notion is meant to make sense of the following construction. I am now  taking a long thin strip of paper, a glue stick and glue the ends of the strip together. Do you know what we got? Aha, somebody in the class already knows about the Moebius strip. Can this construction be described mathematically? The quotient topology is the answer to this puzzle.  
OK, we are done with the introduction. Now, as for the definition of a topological space, let $X$ be a set ... 
A: Focus on the closure of a set as the primary intuitive driven motivation. 

Class, we want to describe the the concept of limit points. As you pursue your studies in mathematics, there will be numerous situations when you will need to know how to describe and analyze 'points' that are 'close' to each other. It is amazing and our good fortune that some abstract set theory machinery, known as topology, has been developed. By turning the crank there you can apply the results to numerous more concrete objects of your attention.
We will begin with two examples:
Example 1: Consider the unit 'open' disc U of points
$U: z \in \mathbb{C}$ with $z < 1$ 
Notice how every point there is also contained in a smaller disc contained in the bigger disc..
Now what do you think the limit points are?
Class discussion: interior points, boundary, closure of the closure, etc.
Example 2: The limit points of $\mathbb{Q}$ in $\mathbb{R}$ is all $\mathbb{R}$. The Rational Numbers are said to be 'dense' in the Real Number Line. In this instance, the real numbers that are limit points of the subset $\mathbb{Q}$, but are not rational numbers. are said to be irrational real numbers.
Class discussion: The real numbers are a complete topological space, etc.
