Topology

Requisites

• MATH20122 - Metric Spaces (Recommended)

Aims

This lecture course unit aims to introduce students to the basic concepts of topological spaces and continuous functions, and to illustrate the techniques of algebraic topology by means of the fundamental group.

Overview

This course unit is concerned with the study of topological spaces and their structure-preserving functions (continuous functions). Topological methods underpin a great deal of present day mathematics and theoretical physics. Topological spaces are sets which have sufficient structure so that the notion of continuity may be defined for functions between topological spaces. This structure is not defined in terms of a distance function but in terms of certain subsets known as open subsets which are required to satisfy certain basic properties. Continuous functions may stretch or bend a space and so two spaces are considered to be topologically equivalent if one can be obtained from the other by by stretching and bending: for this reason topology is sometimes called rubber sheet geometry.

The first half of the course unit introduces the basic definitions and standard examples of topological spaces as well as various types of topological spaces with good properties: pathconnected spaces, compact spaces and Hausdorff spaces . The second half introduces the fundamental group and gives some standard applications of the fundamental group of a circle.

Learning outcomes

On successful completion of this course unit students will be able to:

• prove that certain subsets of Euclidean space are topologically equivalent by constructing a concrete homeomorphism,
• define the notions of path-connectedness and path-components and apply them to distinguish subsets of Euclidean space up to topological equivalence,
• decide whether a collection of subsets of a set determines a topology and whether a map between topological spaces is continuous,
• define the subspace topology, the product topology and the quotient topology, prove their universal properties and apply them to construct continuous maps,
• recognise whether or not a topological space is compact or Hausdorff and state the basic properties of compact and Hausdorff spaces and their proofs,
• define the fundamental group and use it to distinguish topological spaces, apply the functorial properties to find obstructions for the existence of particular continuous maps,
• calculate the fundamental group of the circle and of product spaces.

Assessment methods

• Other - 15%
• Written exam - 85%

Assessment Further Information

• Mid-semester coursework: weighting 15%
• End of semester examination: two hours weighting 85%

Syllabus

1. Topological equivalence: the topological equivalence of subsets of Euclidean spaces,
   path-connected sets and distinguishing subsets of Euclidean spaces using the cut point
   principle. Standard applications of path-connectedness such as the Pancake Theorem.
   [3 lectures]
2. Topological spaces: definition of a topology on a set, a topological space and a
   continuous function between topological spaces; closed subsets of a topological space; a
   basis for a topology. [2]
3. Topological constructions: subspaces, product spaces, quotient spaces; definitions and
   basic properties; standard examples including the cylinder, the torus, the Möbius band,
   the projective plane and the Klein bottle. [5]
4. Compactness: open coverings and subcoverings, definition of a compact subset of a
   topological space; basic properties of compact subsets; compact subsets in Euclidean
   spaces (the Heine-Borel Theorem). [2]
5. Hausdorff spaces: definition and basic properties of Hausdorff spaces; a continuous
   bijection from a compact space to a Hausdorff space is a homeomorphism, quotient
   spaces of compact Hausdorff spaces. [2]
6. The fundamental group: equivalent paths, the algebra of paths, definition of the
   fundamental group and dependence on the base point. [3]
7. The fundamental group of the circle: the path lifting theorem for the standard cover
   of the circle, the degree of a loop in the circle, the fundamental group of the circle,
   standard applications: the Brouwer Non-Retraction Theorem, the Brouwer Fixed Point
   Theorem, the Fundamental Theorem of Algebra, the Hairy Ball Theorem. [5]

Recommended reading

The first three of the following books contains most of the material in the course with the third a little more advanced than the first two. The fourth book contains most of material in the first half of the course and relates topological spaces to metric spaces.

• M. A. Armstrong. Basic Topology, Springer 1997.
• C. Kosniowski, A First Course in Algebraic Topology, Cambridge University Press 1980.
• J. R. Munkres, Topology, Prentice-Hall 2000 (second edition).
• W. A. Sutherland, Introduction to Metric and Topological Spaces, Oxford University Press 2009 (second edition).

Feedback methods

Feedback tutorials will provide an opportunity for students' work to be discussed and provide feedback on their understanding.  Coursework or in-class tests (where applicable) also provide an opportunity for students to receive feedback.  Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer's office hour.

Study hours

• Lectures - 22 hours
• Tutorials - 11 hours
• Independent study hours - 67 hours

Teaching staff