Metrische Ruimten - 2009-2010

The RuG page with information on the course is here.

The final grade will be determined by the final exam grade f and the homework grade h as max(f, 0.2h+0.8f ).

Lessons

In the following, section numbers are given in the form Sec. 1.1 / 4.1 where the first numbers refers to the 1975 edition and the second to the 2009 edition of the book ‘Introduction to Metric and Topological Spaces’ by W.A. Sutherland.

This information appears also in Nestor under “Course Information”.

Lesson 1 (08/02/2010)

Sec. 1.1 / 4.1 Real numbers (definition of sup, inf, their properties, and the completeness axiom). Sec. 1.2 / 4.2 Real sequences (definition of convergence, Cauchy sequences). Sec. 1.3 / 4.3 Limits of functions (definition of limit). Sec. 1.4 / 4.4 Continuity (definition of continuous functions in R). Sec. 2.1 / 5.1 Motivation and definition of metric spaces (also definition of continuity. Sec. 2.2 / 5.2 Examples of metric spaces (Rⁿ with the d₁, d₂, d_p, and d_∞ metrics-no proofs, discrete metric).

Lesson 2 (11/02/2010)

Sec. 2.2 / 5.2 Examples of metric spaces (Proofs that d₁, d₂, and d_∞ on Rⁿ are indeed metrics, definitions and proofs for the sup-metric and L¹ metric in function spaces).

Lesson 3 (15/02/2010)

Sec. 2.2 / 5.2 Examples of metric spaces (Subspace and product space metrics). Sec. 2.2 / 5.4 Bounded sets in metric spaces. Sec. 2.3 / 5.5, 5.6 Open balls and open sets in metric spaces (Definitions of open balls and open sets, examples, characterization of continuity with respect to open sets, definition of closed sets: a subset S is closed in a metric space A if A-S is open in A).

Lesson 4 (18/02/2010)

Sec. 2.3 / 5.6 Open balls and open sets in metric spaces (The union and finite intersection of open sets is open). Sec. 2.4 / 6.7 Equivalent metrics (Definition of topological and Lipschitz equivalence, characterization of topological equivalence in terms of open sets, examples of equivalent and non-equivalent metrics).

Lesson 5 (22/02/2010)

Sec. 3.1 / 7.1, 7.2, 8.1 Topological spaces, examples, and continuity in topological spaces. Sec. 3.2 / 8.3 Bases of topological spaces.

Lesson 6 (25/02/2010)

Sec. 3.6 / 8.2 Homeomorphisms. Sec. 3.4 / 10.1 Subspaces. Sec. 3.5 / 10.2 Products.

Lesson 7 (01/03/2010)

Sec. 3.6 / 10.3 Graphs. Sec. 3.7 / 6.1, 6.2, 6.3, 9 Closed sets, limit points, closures (in topological and metric spaces).

Lesson 8 (04/03/2010)

Sec. 3.7 / 6.4, 6.5, 9 Properties of closures. Interiors, boundaries. Sec. 4.1, 4.2 / 11.1, 11.2 Hausdorff spaces.

Lesson 9 (10/03/2010)

Sec. 6.1, 6.2 / 12.1, 12.2, 12.5 Connected spaces. Connectedness and homeomorphisms.

Lesson 10 (11/03/2010)

Sec. 6.3 / 12.3 Path-connectedness. Sec. 6.4 / 12.4 Comparison of connectedness and path-connectedness. Sec. 5.2 / 13.2 Definition of compactness. Sec. 5.3 / 13.3 Compactness of closed bounded intervals [a,b].

Lesson 11 (15/03/2010)

Sec. 5.4 / 13.4 Properties of compact spaces. Sec. 5.5 / 13.5 Continuous maps on compact spaces. Sec. 5.6 / 13.6 Compactness of subspaces and product spaces. Sec. 5.7 / 13.7 Compact subspaces of Rⁿ. Sec. 5.9 / 13.9 An inverse function theorem.

Lesson 12 (18/03/2010)

Sec. 7.2 / 14.2 Sequential compactness.

Lesson 13 (22/03/2010)

Sec. 9.1, 9.2 / 17.1 Complete metric spaces.

Lesson 14 (29/03/2010)

Sec. 9.3, 9.4 / 17.2, 17.3 Banach's fixed point theorem. Sec. 8.1, 8.2 / 16.1, 16.2 Uniform convergence.

Lesson 15 (01/04/2010)

Sec. 8.3, 8.4, 8.5 / 16.3, 16.4, 16.5 Completeness of the set of bounded maps, and subsets of continuous functions. Sec. 9.4 / 17.4 Applications of Banach's fixed point theorem.