Computer Science and Mathematics (3 Years) [BSc]
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 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 structurepreserving 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 pathconnectedness and pathcomponents 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  20%
 Written exam  80%
Assessment Further Information
 Midsemester coursework: weighting 20%
 End of semester examination: two hours weighting 80%
Syllabus

Topological equivalence: the topological equivalence of subsets of Euclidean spaces,
pathconnected sets and distinguishing subsets of Euclidean spaces using the cut point
principle. Standard applications of pathconnectedness such as the Pancake Theorem.
[3 lectures] 
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] 
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] 
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 HeineBorel Theorem). [2] 
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] 
The fundamental group: equivalent paths, the algebra of paths, definition of the
fundamental group and dependence on the base point. [3] 
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 NonRetraction 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, PrenticeHall 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 inclass 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