MA366 GENERAL TOPOLOGY (3 Cr.)

COURSE DESCRIPTION

Prerequisite: MA 211 and MA 265

General Introduction and Goals

General topology is the study of abstract topological spaces and continuous maps between such spaces. As such, it serves as a foundation for geometric topology, manifold theory, differential topology and algebraic topology as well as for analysis. The theory of metric spaces, both in their own right and as topological spaces, is usually included within the scope of introductory courses in general topology.

General Topology may be viewed as a general theory of convergence and as such is of basic importance in most mathematical disciplines. For example, a grasp of general topology is fundamental to the understanding of the real number line, higher dimensional Euclidean spaces, the function spaces of analysis, and many of the deeper theorems of Boolean algebra and mathematical logic.

The two pillars of general topology are compact spaces and metric spaces and continuous mappings between these spaces. As such, the goal of the course is to study compact topological spaces and metric spaces and continuous maps between these spaces. An additional goal is to develop an understanding of the role of topology in other branches of mathematics. Within an introductory course not too much can be covered, but some non-trivial application should at least be touched upon. For example, one of these topics might be considered: the contraction principle and its application to differential equations; the Hausdorff metric and its application to fractal geometry;or the process of completing a metric space and its application to an abstract understanding of the Lebesgue integral.

Course Content

The course content will vary with the instructor. However, metric spaces and compact spaces and continuous maps on these spaces will always be of central importance. One possibility, in broad outline, is given below.

1. The topology of the real line and the Euclidean plane.
   • Convergence and continuity
   • Subspaces
   • Connectedness
   • Compactness
2. Metric spaces.
   • Convergence and continuity
   • Subspaces
   • Connectedness
   • Compact sets and spaces
   • Normality
3. Topological spaces.
   • Continuous maps
   • Subspaces
   • First countable spaces and sequential convergence
   • Hausdorff spaces
   • Compact spaces and sets
   • Connectedness
   • Normal spaces
4. Product spaces and quotient spaces
   • Quotient maps and quotient spaces
   • Open, closed and perfect maps
   • Finite and countable products
   • General products
   • The countable Tychonoff Theorem
   • A universal second countable metric space
5. A final topic and application.

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