# 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.

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:

1. Continuity of functions/mappings.

2. Connectivity.

3. 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 …