Entry requirements

Linear Algebra II and Several Variable Calculus.

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.

On completion of the course, the student should be able to:

• 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 examination at the end of the course combined with assignments during the course.

If there are special reasons for doing so, an examiner may make an exception from the method of assessment indicated and allow a student to be assessed by another method. An example of special reasons might be a certificate regarding special pedagogical support from the disability coordinator of the university.

Other directives

The course cannot be included in passing degree together with the course Topology I.