MX3036: METRIC AND TOPOLOGICAL SPACES (2018-2019)

• Metric spaces and topological spaces.
• Compactness, connectedness.
• Subspace and product space topology.
• Complete metric spaces.

Syllabus

• Metric spaces
• Continuity and convergence in metric spaces
• Open and closed sets; interior and closure
• Products and subspaces of metric spaces
• Connectedness and path-connectedness
• Compactness
• Completeness
• Topological spaces
• One-point compactification; quotient spaces

Course Aims

The aim of the course is to introduce the basic concepts of metric spaces and their associated topology, and to apply the ideas to Euclidean space and other examples.

Learning Objectives

By the end of the course, students should be familiar with the basic topological aspects of metric spaces (open and closed sets, continuous functions, connectedness and compactness), and with the metric notions of uniform continuity and completeness. By the end of the revision period, students should be able to

• state and illustrate the definitions of the concepts introduced in the course,

• state the theorems of the course, to explain their significance, and to give examples to indicate the role of the hypotheses,

• demonstrate knowledge and understanding of proof techniques used in the course, and

• use the methods and results of the course to solve problems at levels similar to those seen in the course.

Information on contact teaching time is available from the course guide.

More Information about Week Numbers

1st Attempt: 1 two-hour written examination (80%); in-course assessment (20%). Resit: 1 two-hour examination (maximum of 100% resit and 80% resit with 20% in-course assessment).

Informal assessment of weekly homework through discussions in tutorials.

In-course assignments will normally be marked within one week and feedback provided to students in tutorials. Students will be invited to contact Course Coordinators for feedback on the final examination.