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Online course materials for MATH31052

Topology


Unit code: MATH31052
Credit Rating: 10
Unit level: Level 3
Teaching period(s): Semester 2
Offered by School of Mathematics
Available as a free choice unit?: N

Requisites

Prerequisite

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 homeomorphic (topologically
    equivalent) by constructing a homeomorphism;
  • understand the notions of path-connectedness and path-component and be familiar with
    some standard applications;
  • determine whether a collection of subsets of a set determines a topology;
  • use the definitions of the subspace topology, the product topology and the quotient
    topology, understand the use and proof of their universal properties and be familiar with
    standard examples such as topological surfaces;
  • recognize whether or not a subset of a topological space is compact and be familiar with
    the basic properties of compact subsets and their proofs;
  • recognize whether or not a topological space is Hausdorff and be familiar with the basic
    properties of Hausdorff spaces and their proofs;
  • understand the definition of the fundamental group and its functorial properties;
  • calculate the fundamental group of the circle using its univeral covering space and have
    seen some standard applications.

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

Hendrik Suess - Unit coordinator

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