# Unit information: Metric Spaces in 2022/23

Unit name Metric Spaces MATH20006 20 I/5 Teaching Block 1 (weeks 1 - 12) Dr. Viveka Erlandsson Not open MATH10011 Analysis and MATH10010 Introduction to Proofs and Group Theory None None School of Mathematics Faculty of Science

## Unit Information

Lecturers: Viveka Erlandsson

Unit Aims

To introduce the notion of metric spaces and extend concepts familiar from Analyis 1 about the real numbers and real valued functions, such as convergence and continuity, to the more general setting of these spaces.

Unit Description

The theory of metric spaces is a fundamental area of pure mathematics and is an entry gate into more abstract areas such as pointset topology. A metric space is a set together with a notion of a distance between points in the set, and generalises familiar spaces such as the real numbers or the plane with the usual notion of (Euclidean) distance. This allows us to put ideas from analysis in a new, more abstract, light. After defining metric spaces and seeing plenty of examples, we will discuss their properties, many of which will be familiar from analysis. Central concepts will be that of convergence of sequences and continuity of maps, and we will also define other notions such as open and closed sets, compact and complete sets, and connectedness.

Relation to Other Units

This unit is a member of a sequence of analysis units at levels 5, 6 and 7. It is a prerequisite for Measure Theory and Integration, Advanced Topics in Analysis, Functional Analysis, and Geometry of Manifolds.

Topics covered will include: Definition and examples of metric spaces; convergence in metric spaces; maps between metric spaces and continuity; open and closed sets; the notions of complete, compact, and connected metric spaces and subsets; the contraction mapping theorem and its applications. The course will end with a (non-examinable) brief introduction to a related, but more general, type of spaces: topological spaces.

## Your learning on this unit

At the end of the course the student should know and understand the basic definitions and theorems (and their proofs). They should be able to solve routine problems similar to assigned exercises as well as to apply the ideas to unseen situations.

## How you will learn

The unit will be taught through a combination of

• Face-to-face lectures
• asynchronous online materials, including narrated presentations and worked examples
• guided asynchronous independent activities such as problem sheets and/or other exercises
• synchronous weekly group problem/example classes, workshops and/or tutorials
• synchronous weekly group tutorials
• synchronous weekly office hours

## How you will be assessed

90% Timed, open-book examination 10% Coursework

Raw scores on the examinations will be determined according to the marking scheme written on the examination paper. The marking scheme, indicating the maximum score per question, is a guide to the relative weighting of the questions. Raw scores are moderated as described in the Undergraduate Handbook.

If you fail this unit and are required to resit, reassessment is by a written examination in the August/September Resit and Supplementary exam period.

## Resources

If this unit has a Resource List, you will normally find a link to it in the Blackboard area for the unit. Sometimes there will be a separate link for each weekly topic.

If you are unable to access a list through Blackboard, you can also find it via the Resource Lists homepage. Search for the list by the unit name or code (e.g. MATH20006).

How much time the unit requires
Each credit equates to 10 hours of total student input. For example a 20 credit unit will take you 200 hours of study to complete. Your total learning time is made up of contact time, directed learning tasks, independent learning and assessment activity.