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M4051 Topology
Aims:
Topology is the geometry of continuity, and as such it is involved in almost all parts of mathematics. The aim of this unit is to explain the basic concepts of topology, with the help of examples from geometry, analysis and algebra, and to see how topology interacts with geometric, algebraic and analytic properties.
An example from classical geometry is the Euler polyhedron formula, which shows that the quantity
(number of vertices) - (number of edges) + (number of faces)
is independent of the way a surface is divided into polygonal regions. A related example in differential geometry is the Gauss-Bonnet theorem, which shows that the total curvature of a surface depends only on the number of "holes'' in it.
Syllabus:
Open, closed and compact sets. Continuous mappings and homeomorphisms. Curves, surfaces and other manifolds, in particular spheres and projective spaces. From the Euler polyhedron formula to homology. The classification of surfaces: genus and orientability. Fundamental group, quotient spaces and coverings. Genus and the Gauss-Bonnet theorem.
Prerequisites:
MTH2111 / MTH3111 Mathematical Structures or MTH3132 Analysis and Geometry
References:
Armstrong, M. A., Basic topology, McGraw-Hill, 1979.
Lecturer: John Stillwell
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