
PMA4003 Topology (1^{st} semester)
Prerequisite: PMA3017 Metric and Normed Spaces, and PMA3014 Set Theory
Topology (rather like Algebra or Analysis) is not so much a single branch of mathematics but a loose confederation of subject areas differing widely in their origins, techniques and motivation but united by sharing a common core of basic concepts and constructions. Problems of a topological nature include: how can we describe and classify knots? how can we describe and classify surfaces? to what extent is it possible to extend the ideas of analysis into sets that don't have metrics defined on them? what can be meant by saying that two objects are “fundamentally the same shape”, and how do we decide whether they are or not? what ‘models’ are available to describe certain aspects of theoretical computer science? Rather than attempting to supply answers to any such major questions, this module will concentrate on developing enough of the ‘common core’ to allow students to begin to appreciate how such issues can be tackled topologically.
Ordered Sets: Partial order, maximal and maximum, minimal and minimum, upper and lower bounds, infima and suprema, lattices.
Topological Spaces: Topologies, open set, closed set, closure, neighbourhood, base, subbase, lattice of topologies, subspaces.
Continuity and Sequential Convergence: Continuous functions between topological spaces, composites, homeomorphism, topological invariants, contractive and expansive invariants. Convergence of a sequence in a topological space; the inadequacy of sequential convergence in topological spaces, contrasting with the adequacy of sequential convergence in metric spaces. Firstcountable spaces, separable spaces, secondcountable spaces.
Separation Axioms: T_{0}spaces, T_{1}spaces, Hausdorff or T_{2}spaces, regular or T_{3}spaces, normal or T_{4}spaces. Every secondcountable T_{3}space is metrizable and separable.
Compactness: Open covers, compact subsets, KCspaces, a compact T_{2}space is T_{4}, maximal compact and minimal T_{2}spaces.
Connectedness: Connected spaces, connected subsets, components, totally disconnected spaces, locally connected spaces.
Assessment
Exam 70% 2x Mini Project 30% (15% each)
Textbooks
The libraries contain several texts on topology which include sections relevant to this module, but be aware that a lot of them are written at postgraduate level. More accessible are:
Mansfield, M. J., Introduction to Topology (Van Nostrand).
Mendelson, B., Introduction to Topology (Allyn & Bacon).
Moore, T. O., Elementary General Topology (Prentice Hall).
Simmons, G. F., Introduction to Topology and Modern Analysis (McGrawHill).
Lipschutz, S., General Topology (Schaum).