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MA341 Metric Spaces
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Metric spaces: examples of metric spaces; convergence in metric spaces;
pointwise, uniform and mean convergence; continuity; open and closed sets;
closure, interior and boundary; compactness in metric spaces; the Bolzano-Weierstrass
theorem
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Completeness: Rn and C[a,b]; contractions; the
fixed point theorem; applications to differential equations etc
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Fractal geometry: the space of fractals; the Hausdorff metric; iterated
function systems; algorithms for generating fractals; fractal dimension
Texts
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A.N. Kolmogorov & S.V. Fomin, "Real Analysis" (Dover)
MA342 Topology
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Topological spaces: examples; continuity and convergence; subspaces, quotients
and product spaces
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Connectedness and path connectedness: components; totally disconnected
spaces
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Compactedness and its applications: the Heine-Borel theorem; compactness
of subspaces and product spaces; compactness and sequential compactness
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Convergence: the Hausdorff and other separation properties; inadequacy
of sequences; nets; filters and ultrafilters
Texts
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S. Lipschutz, "General Topology" (Schaum Outline)
References
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A. Armstrong, "Basic Topology" (Springer)
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S. Barr, "Experiments in Topology" (Murray)
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M. Barnsley, "Fractals Everywhere" (Academic Press)
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N. Bourbaki, "General Topology" (Hermann)
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V. Bryant, "Metric Spaces, Iteration and Applications" (Cambridge)
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E. Copson, "Metric Spaces" (Cambridge)
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K. Falconer, "Fractal Geometry" (Wiley)
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G. Francis, "A Topological Picturebook" (Springer)
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V. Mendelson, "Introduction to Topology" (Alyn & Bacon)
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J. Munkres, "Topology, A First Course" (Prentice Hall)
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I.M. Singer & J.A. Thorpe, "Lecture Notes on Elementary Topology and
Geometry" (Springer)
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W.A. Sutherland, "Introduction to Metric and Topological Spaces" (Oxford)
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S. Willard, "General Topology" (Addison Wesley)
MA343 Group Theory (part one)
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Group axioms, cyclic groups, permutation groups.
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Normal subghroups, homomorphisms, isomorphism theorems.
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Permutation groups, isomorphism theorems, linear groups
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Direct and semidirect products
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Finitely generated Abelian groups
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Automorphisms, groups of automorphisms
Back to 4th Year Syllabus
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