Module Catalogues, Xi'an Jiaotong-Liverpool University

Module Code: MTH224
Module Title: Metric Spaces
Module Level: Level 2
Module Credits: 5.00
Semester: SEM2
Originating Department: Mathematical Sciences
Pre-requisites: MTH117

 Aims Metric spaces are sets equipped with a notion of distance. This module examines how important ideas and concepts of real analysis can be extended to the more general setting of metric spaces. This module provides a foundation for further study in a wide range of mathematical topics, including analysis, topology, measure theory, geometry and dynamical systems.
 Learning outcomes A Recognise whether a given structure forms a metric space.B Work effectively with subsets of a metric space and determine the diameter, boundary, interior or closure.C Prove results concerning continuous maps between metric spaces.D Demonstrate whether or not a given metric space is connected, compact or complete.E Use the Banach contraction mapping theorem to identify fixed points.F Solve mathematical problems using the formalism of metric spaces.
 Method of teaching and learning This module will be delivered mainly through a combination of formal lectures and tutorials.
 Syllabus 1. Metric spaces: axioms, survey of examples in different contexts, metrics from norms, metrics on subspaces, metrics on products, isometries.2. Distances: diameter, isolated points, distance between sets.3. Open sets, closed sets, boundary points, closure, interior, dense and nowhere dense sets.4. Continuity, sequential continuity, topological approach to continuity, homeomorphism, uniform continuity, Lipschitz continuity.5. Completeness, Cauchy sequences, Cantor intersection theorem, completeness of C([0,1]), contractions, contraction mapping theorem, applications, Picard iteration.6. Connectedness, path connectedness, connectedness and continuity.7. Compactness, open covers, equivalence of sequential compactness, total boundedness, compactness and continuity, equivalence of norms on Rn.8. Applications of the theory of metric spaces:Fractals - Cantor set, Hausdorff metric, iterated function systems.Data science - clustering algorithms.
Delivery Hours
 Lectures Seminars Tutorials Lab/Prcaticals Fieldwork / Placement Other(Private study) Total Hours/Semester 39 13 98 150

## Assessment

 Sequence Method % of Final Mark 1 Coursework 1 10.00 2 Coursework 2 10.00 3 Final Exam 80.00
 Module Catalogue generated from SITS CUT-OFF: 6/5/2020 9:22:57 PM