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MX3036: METRIC AND TOPOLOGICAL SPACES (2017-2018)

Last modified: 23 Aug 2017 15:34


Course Overview

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.

An excellent introduction to "serious mathematics" based on the usual geometry of the n dimensional spaces.



Course Details

Study Type Undergraduate Level 3
Session First Sub Session Credit Points 15 credits (7.5 ECTS credits)
Campus None. Sustained Study No
Co-ordinators
  • Dr Mark Grant

Qualification Prerequisites

  • Either Programme Level 3 or Programme Level 4

What courses & programmes must have been taken before this course?

What other courses must be taken with this course?

None.

What courses cannot be taken with this course?

Are there a limited number of places available?

No

Course Description

  • 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

Further Information & Notes

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.

Degree Programmes for which this Course is Prescribed

  • BSc Mathematics
  • BSc Mathematics with Gaelic
  • MA Mathematics
  • Mathematics Major

Contact Teaching Time

33 hours

This is the total time spent in lectures, tutorials and other class teaching.

Teaching Breakdown


Assessment

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). Only the marks obtained on first sitting can be used for Honours classification.

Formative Assessment

Informal assessment of weekly homework through discussions in tutorials.

Feedback

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.

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