Entry requirements

120 credit points and 90 credit points Mathematics

Learning outcomes

The course aims at strengthening and generalizing results that the student has already learnt from previous calculus courses, providing her or him with an adequate language for advanced studies of mathematics, and developing skills in working with abstract concepts whose meaning are defined by various sets of axioms. It is important that the student learns how to apply the general and abstract concepts and results presented during the course in concrete cases.

In order to pass the course (grade 3) the student should

• master the various topological and metrical concepts that are introduced in the course – know their definitions and how to use them in concrete situations. The most important of these concepts are: open and closed sets, closure, interior, boundary, dense sets, the canonical topology in a metric space, the euclidean topology, topological subspace, continuous map, homeomorphism, connectedness, compactness, separability, homotopy, homotopy of paths and the fundamental group;
• be able to give an account of various set theoretic and topological constructions, such as products and factorisation of topological spaces;
• be able to describe the hereditary under continuous maps and product formation of various topological properties.

Content

Topological spaces: basic definitions, subspaces. Metric spaces: metric topology, metrisability.

Continuous maps. Homeomorphisms. Topological embeddings. Connectedness and pathwise connectedness. Separation axioms. First and the second axiom of countability. Compactness. Sequential compactness. Products of topological spaces. The quotient topologi. Pasting of topological spaces. Real and complex projective spaces. Homotopy, homotopy of paths, the fundamental group, covering spaces.

Instruction

Lectures and problem solving sessions.

Assessment

Written examination at the end of the course combined with assignments given during the course.