This course provides a sound grounding in the basic properties of metric and topological spaces, and in particular extends the concept of continuity to metric spaces and then to topological spaces

34 x 1hr lectures and 12 x 1hr tutorials in a semester.

Geometry and Topology I 86PL

Geometry and Topology II 86PM

90% Examination, 10% Coursework.

The main aim of this course is to provide a sound grounding in the basic properties of metric and topological spaces. In particular, the concept of continuity which was introduced rigorously in Level 2 for real functions is extended to metric spaces and then to topological spaces. Geometric applications including topological groups and group actions and an introduction to surfaces are also included.

By the end of this course students will be able to:

■ show that a function is or is not a metric;

■ show that a set in a metric space is or is not open and/or closed;

■ show that a function between metric spaces is or is not continuous;

■ show that a sequence in a metric space is or is not convergent;

■ show that a metric space is or is not complete;

■ verify that a given family of sets is a topology;

■ show that two topological spaces are (or are not) homeomorphic;

■ explain what it means for a topological space to be compact, connected, path-connected, Hausdorff, first countable, second countable, and verify that each of these is a topological property;

■ explain the ideas associated with the product topology (two factor spaces) and the quotient topology;

■ verify that certain groups are topological groups;

■ identify the orbit spaces of some group actions;

■ explain the ideas associated with surfaces and their classification;

■ prove the main theorems in the course, and use these results to solve other problems.