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Metric Spaces (MATH 20200)

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Administrative Information

  1. Unit number and title: MATH 20200 Metric Spaces
  2. Level: I/5
  3. Credit point value: 20 credit points
  4. Year: 12/13
  5. First Given in this form: 1996 (as Analysis 2)
  6. Unit Organiser: Yuri Netrusov
  7. Lecturer: Yuri Netrusov
  8. Teaching block: 1
  9. Prerequisites: MATH11006 Analysis 1 and MATH 11521 Further Topics in Analysis

Unit aims

To introduce metric spaces and to extend some theorems about convergence and continuity in the case of sequences of real numbers and real-valued functions of one real variable.

General Description of the Unit

This course generalizes some theorems about convergence and continuity of functions from the Level 4 unit  Analysis 1, and develops a theory of convergence in Rn and more generally in any metric space. Topics will include basic topology (open, closed, compact, connected sets), continuity of functions, completeness, the contraction mapping theorem and applications, compactness 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 & Integration and for Functional Analysis.

Teaching Methods

A standard lecture course of 33 lectures with 8 - 10 problem classes.

Learning Objectives

At the end of the course the student should know and understand the definitions and theorems (and their proofs) and should be able to use the ideas of the course in unseen situations.

Assessment Methods

The final assessment mark for Metric Spaces is calculated from a 2½-hour written examination in April. The paper consists of FIVE questions. A candidate's FOUR best answers will be used for assessment. Calculators are NOT permitted.

Award of Credit Points

To be awarded the credit points for this unit you must normally pass the unit, i.e. you must achieve an assessment mark of at least 40.

Transferable Skills

Assimilation of abstract ideas and reasoning in an abstract context. Setting out a sustained argument in a form comprehensible to others.

Texts

J.C. Burkill & H. Burkill, A second course in mathematical analysis, Cambridge University Press, Cambridge

I. Kaplansky, Set theory and metric spaces, Chelsea Publishing Company, New York.

W. Rudin, Principles of Mathematical Analysis, McGraw-Hill.

W. A. Sutherland, Introduction to metric and topological spaces, Clarendon Press, Oxford.

Syllabus

Metric spaces (definition, examples, open sets, closed sets, interior, closure, limit points, equivalent metrics, product metrics).

Completeness (limits, continuity, Cauchy sequence, complete sets, isometries, completion of a metric space, contraction mapping theorem, existence and uniqueness of ordinary differential equations).

Compactness (definition, examples, continuous functions, uniform continuity, Heine-Borel theorem, criteria for compactness).

Connectedness (definition, examples, Rn, components, continuous functions, path connectedness).

Uniform convergence.