M3P1/M4P1

METRIC AND TOPOLOGICAL SPACES

Syllabus

In metric and topological spaces ideas associated with convergence and continuity are generalised giving insight and analytical background. Metric and topological spaces: review of convergence and continuity. Metric spaces. Examples (Euclidean spaces, function spaces; uniform convergence). The open sets in a metric space; equivalent metrics. Formulation of convergence and continuity in terms of open sets: topological spaces. Subspaces. Hausdorff spaces. Connected and pathconnected spaces; equivalence of these notions for open sets in Rn. Compact spaces; determination of compact subspaces of Rn . Completeness in metric spaces. Relationship between compactness and completeness.

Basic Bibliography

W.A. Sutherland, Introduction to Metric and Topological Space (Paperback)
A.N. Kolmogorov, S.V. Fomin, and R.A. Silverman , Introductory Real Analysis

PROBLEM SETS

http://planetmath.org/

Some Knots Parametrization http://www2.carthage.edu/~trautwn/hparam.html

Knot ZOO http://www.pims.math.ca/knotplot/zoo/

Knot History http://www.maths.ed.ac.uk/~aar/knots/index.htm

* Dimension http://en.wikipedia.org/wiki/Topological_dimension

* (Fractal Dimension) http://en.wikipedia.org/wiki/List_of_fractals_by_Hausdorff_dimension

* Hausdorff Measure

* Metric Entropy

http://www.math.uni-hamburg.de/home/gunesch/entropy.html

* Topological Degree

* Fixed Point Theory

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* TopologicalZoo_ http://www.geom.uiuc.edu/zoo/

* http://www.kleinbottle.com/whats_a_klein_bottle.htm

* Klein Bottle_ http://plus.maths.org/issue26/features/mathart/index-gifd.html

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PG Studies at IC: MSc & PhD

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