TOPOLOGY (code: 353) SEMESTER: FIRST
CONTACT: DR T VORONOV (M/P5) CREDIT RATING: 10
Aims: To introduce students to the basic concepts of topological spaces and continuous maps, get them acquainted with various examples of spaces and maps, and to give an introduction to algebraic-topological ideas.
Intended Learning Outcomes: On successful completion of the course students will:
  • Have acquired solid knowledge of main properties of topological spaces and continuous maps
  • Have studied a variety of examples of topological spaces
  • Be able to calculate the Euler characteristic for simple complexes and spaces
  • Appreciate the topological classification of compact connected surfaces
Pre-requisites: 153, 211, 251
Dependent Courses: MT4522.
Course Description: This course is concerned with the study of topological spaces and their structure preserving maps (continuous maps). Topological methods and ideas are universal in modern mathematics. A topological space is a type of geometric object that does not necessarily have a distance function associated with it but, via the notion of open sets (introduced by axiomatic properties), it encodes the rigorous approach to the idea of "continuity" and "nearness". Many different examples of topological spaces will be examined, including a class of topological spaces called topological manifolds (spaces that locally look like a Euclidean space), and in particular 2-dimensional topological manifolds, called "surfaces". Examples of surfaces are the sphere, R2, torus, Klein bottle. The course concludes with an introduction to combinatorial topology (which considers topological spaces by means of simplicial complexes and triangulations). We discuss the classification theorem for closed surfaces.
Teaching Modes: 3 classes (lectures and examples) per week
Private Study: 5 hours per week
Recommended Texts: M A Armstrong, Basic Topology, McGraw-Hill.
Assessment Methods: Coursework: 20%
Coursework Mode: Multiple choice test in Week 5.
Examination: 80%
Examination is of 2 hours duration at the end of the FIRST semester.
No of weeks: Syllabus
1 Topological properties: an idea. Definition of a topological space. Continuous maps and homeomorphisms. Examples.
2 Constructions of new topological spaces. Induced topology and subspaces. Coinduced topology and identification spaces. Product spaces. Examples.
3 More on closed sets. Separation properties, Hausdorff spaces. Compactness. Connectedness and path-connectedness.
2 Topological manifolds. Examples. Surfaces. Orientability. The classification theorem for closed surfaces (statement).
3 Simplicial complexes. Euler characteristic. The excision formula. Examples of calculation. Application to surfaces.

 

     Last revised 27 September/ 10 October 2005