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.
We will use Basic Topology by M. A. Armstrong, Springer-Verlag 1994 as our text. Click here for the table of contents and other information about the text.
Grades in the course will be based on homeworks, class participation, quizes and a take-home final.
Math. 655 is the first of a year long sequence. In the followup courses Math. 656 and 657, we will study the fundamental group and its applications, and homology respectively.