Entry requirements

120 credit points including at least 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.

In order to pass the course the student should

• master the various topological and metrical concepts that are introduced in the course, be able to give an account of their definitions and 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;
• 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 heredity of various topological properties under continuous maps and product formation.

Content

Topological spaces: basic definitions, subspaces. Metric spaces: metric topology, metrisability. Continuous maps. Homeomorphisms. Topological embeddings. Connectedness and pathwise connectedness. Separation axioms. First and second axiom countability. Compactness, sequential compactness. Products of topological spaces. The quotient topology. Pasting of topological spaces. Homotopy of paths.

Instruction

Lectures and problem solving sessions.

Assessment

Written and, possibly, oral examination at the end of the course. Moreover, compulsory assignments may be given during the course.