# What is Topology? ### What is it good for?

1) It is the natural setting for basic notions of analysis like continuity and such fundamental results like:

Intermediate Value Theorem: If f:[a,b] -> R is continuous, then f(x) assumes every value between f(a) and f(b).

Extreme Value Theorem: If f:[a,b] -> R is continuous, then f(x) assumes both a maximum and minimum value

From the viewpoint of topology it is seen that these results are simple manifestations of the very natural notions of connectedness and compactness, respectively, and from this vantage the proofs of these results (as well as their generalizations) become extremely simple, almost obvious.

2) It is the underpinning of modern geometry. Using topological methods we can construct a great variety of interesting geometric objects previously unimagined, and tackle formerly inaccessible questions about geometric objects such as knots and links. We discover that certain seemingly isolated curiousities such as Euler's formula for polyhedra are the tip of an iceberg of a beautiful connection between algebra and geometry.

Math. 655 is a course designed for beginning graduate students and advanced undergraduates. The course introduces the basic notions of topology: continuity, connectedness, compactness, quotient spaces, manifolds.

A good familiarity with the mathematics related to calculus of functions of many real variables is a prerequisite for the course. This includes countably and uncountably infinite sets, equivalence relations, continuity of functions in n real variables and convergence of sequences of points in n-dimensional Euclidean space (at least for n=2 and 3 and being receptive to the idea that these notions generalize to higher n). Some knowledge of linear algebra is helpful in dealing with higher dimensional Euclidean spaces.