Metrische Ruimten - 2008-2009

The RuG page with information on the course is here.

The final grade will be determined exclusively by the final exam.

Lessons

In the following, chapter and page numbers refer to the book ‘Introduction to Metric and Topological Spaces’ by W.A. Sutherland.

Lesson 1 (11/02/2009)

Convergence of real sequences. Cauchy sequences in R. The completeness axiom. Continuity of function in R. Definition of metric spaces and examples. Continuity in metric spaces.

Lesson 2 (13/02/2009)

More examples of metric spaces (function spaces, products, subspaces). Open balls and open sets. Continuity in terms of open sets.

Lesson 3 (18/02/2009)

Topologically and Lipschitz equivalent metrics. Definition of topological spaces. Examples.

Lesson 4 (20/02/2009)

More examples of topological spaces. Homeomorphisms and examples. Bases of topological spaces.

Lesson 5 (25/02/2009)

Subspace topology. Product topology. Closed sets.

Lesson 6 (27/02/2009)

Limit points, closure, interior, boundary, and examples. Quotient spaces.

Lesson 7 (04/03/2009)

Hausdorff spaces. Compactness: definition and examples. The Heine-Borel theorem. Compact subsets of metric spaces are bounded, compact subsets of Hausdorff spaces are closed.

Lesson 8 (06/03/2009)

Continuous images of compact spaces are compact, closed subspaces of compact spaces are compact, compactness and topological product, closed and bounded subspaces of Rⁿ are compact. Uniform continuity, inverse function theorem

Lesson 9 (11/03/2009)

Connected spaces.

Lesson 10 (13/03/2009)

Path-connectedness, comparison of connectedness and path-connectedness. Sequential compactness.

Lesson 11 (18/03/2009)

Equivalence of compactness and sequential compactness (chapter 7).

Lesson 12 (20/03/2009)

Completeness (sections 9.1-9.2). Complete, precompact spaces are compact (section 10.1).

Lesson 13 (25/03/2009)

Uniform convergence of function sequences. Completeness of the space of bounded functions and the subspace of continuous functions.

Lesson 14 (27/03/2009)

The Banach fixed point theorem. The Arzelà-Ascoli theorem.

Lesson 15 (01/04/2009)

Exercises.

Lesson 16 (03/04/2009)

Exercises.