#### Prerequisites & Bulletin Description

#### Course Objectives

The aim of Topology is to introduce the theory of metric spaces and topological spaces. Students are expected to learn how to write, in logical manner, proofs using important theorems and properties of metric spaces and topological spaces.

Students learn to solve problems using the concepts of topology. They present their solutions as rigorous proofs written in correct mathematical English. Students will be able to devise, organize and present brief solutions based on definitions and theorems of topology.

Students who successfully complete this course should be capable of understanding the concept of open and closed sets, the interior, closure and boundary of sets; connected sets, compact sets and continuous functions defined on topological spaces. Students should be able to determine the interior, closure and boundaries of sets, and determine whether a given set is connected or compact. They should be able to determine whether a function defined on a metric or topological space is continuous or not.

Students are expected not only to grasp the concepts of topology and apply them, but also to continue with their overall mathematical development. They will be improving such skills as mathematical writing and the presentation of rigorous logical arguments.

#### Evaluation of Students

Students will be graded on their ability to devise, organize and present in correct mathematical English rigorous solutions to assignments and problems on exams. While instructors may design their own methods of grading student performance these methods must include in-class examinations, graded homework assignments and a final exam.

#### Course Outline

Topics |
Number of Weeks |

Metric Spaces: open and closed sets, interior, closure, boundary of sets, connected sets, compact sets, and continuous functions. | 6 |

Topological Spaces: the same concepts as above, this time in the context of general topological spaces. | 6 |

Special Topics: function spaces, introduction to algebraic topology. | 3 |

#### Textbooks & Software

Munkres, *Topology*

Wade, *An Introduction to Analysis*

Submitted by: Alex Schuster and David Ellis

Date: Apr. 24th, 2003