1. Topological spaces and continuity (1.1 Topological space, open and closed sets, continuous maps, homeomorphisms, and related notions.)

2. Connectedness (2.1 Connected and path-connected spaces, connectedness of intervals, the Euclidean line is not homeomorphic to the Euclidean plane.)

3. Compactness (3.1 Definition of compactness, characterisations for subspaces of Euclidean spaces and metric spaces, applications existence of maxima and minima, Hausdorff spaces.)

4. Quotient topology (4.1 Definition of quotient space, examples especially gluing constructions.)

Learning outcomes

By the end of the course, students should be able to do the following:

• State the definitions of the key notions of point-set-topology, including the definitions of continuous map, connectedness, and compactness.
• Demonstrate their understanding of these key notions by being able to prove abstract statements about them and how they relate to each other.
• Demonstrate their intuition by being able to determine whether concrete topological spaces presented to them have or do not have a specified topological property.
• Construct examples of topological spaces with specified lists of properties, based on other known examples.
• Use their knowledge of topology to solve problems in other areas of mathematics, such as analysis.

Further details

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SCQF Level: 10

Credits: 15