Metrische Ruimten - 2006-2007

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 (14/11/2006)

Covered the main points of Ch. 1 (Review of some real analysis). Defined metric spaces and gave several examples of metric spaces (roughly sections 2.1 and 2.2, pages 19-29).

Lesson 2 (16/11/2006)

Covered the rest of Ch. 2. Defined open balls and open sets in a metric space and gave the characterization of continuity with respect to open sets. Defined topological and Lipschitz equivalence of metrics, and homeomorphisms.

Lesson 3 (21/11/2006)

Defined topological spaces and gave examples. Continuity in topological spaces. Defined the basis of the topology (sections 3.1-3.3).

Lesson 4 (23/11/2006)

Defined the subspace and the product topology, homeomorphisms and closed sets (sections 3.4-3.7, pages 51-61).

Lesson 5 (28/11/2006)

Defined the notions of the limit point of a set, the closure, the interior and the boundary (section 3.7, pages 61-64). Defined Hausdorff spaces (chapter 4, pages 72-73).

Lesson 6 (30/11/2006)

Definition of compact topological spaces. Compact subsets of metric spaces are bounded, compact subsets of Hausdorff spaces are closed. The Heine-Borel theorem (chapter 5, sections 5.1-5.4).

Lesson 7 (5/12/2006)

Continuous images of compact spaces are compact, closed subspaces of compact spaces are compact, compactness and topological product, closed and bounded subspaces of R^n are compact (chapter 5, sections 5.5-5.7).

Lesson 8 (7/12/2006)

Uniform continuity, inverse function theorem (chapter 5, sections 5.8-5.9). Connected spaces (chapter 6, sections 6.1-6.2).

Lesson 9 (12/12/2006)

Connected spaces, path-connectedness, comparison of definitions (chapter 6, sections 6.2-6.4).

Lesson 10 (14/12/2006)

Sequential compactness. Equivalence to compactness (chapter 7).

Lesson 11 (19/12/2006)

Uniform convergence (sections 8.1-8.3).

Lesson 12 (21/12/2006)

Uniform limits of sequences (section 8.4). Complete metric spaces. Banach fixed point theorem (sections 9.1,9.2,9.4).

Lesson 13 (09/01/2007)

Application of the Banach fixed point theorem. Cantor's and Baire's theorem (sections 9.4, 9.5).

Lesson 14 (11/01/2007)

Completeness and precompactness imply compactness (section 10.1). The Arzela-Ascoli theorem (section 10.2).

Lesson 15 (16/01/2007)

Exercises 2.6.5, 3.9.6, 3.9.19, 3.9.25, 3.9.31, 4.3.3, 5.10.1, 5.10.19.

Lesson 16 (18/01/2007)

Exercises 9.6.3, 9.6.19, 10.3.2, 10.3.7(i), 10.3.13.

List of exercises

Ch. 2

4, 8, 10, 11, 19, 26.

Ch. 3

2, 10, 11, 14, 18, 22, 29, 30, 35.

Ch. 4

1, 4.

Ch. 5

3, 5, 6, 7, 10, 14, 25, 34.

Ch. 6

2, 4, 7, 12, 14, 23.

Ch. 7

1, 2, 4, 5, 7, 12.

Ch. 8

1, 2, 3, 7, 8, 9.

Ch. 9

1, 2, 5, 7, 12, 15, 16