Syllabus  MA341/MA342
MA341 Metric Spaces

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 BolzanoWeierstrass
theorem

Completeness: R^{n} and C[a,b]; contractions; the
fixed point theorem; applications to differential equations etc

Fractal geometry: the space of fractals; the Hausdorff metric; iterated
function systems; algorithms for generating fractals; fractal dimension
Texts

A.N. Kolmogorov & S.V. Fomin, "Real Analysis" (Dover)
MA342 Topology

Topological spaces: examples; continuity and convergence; subspaces, quotients
and product spaces

Connectedness and path connectedness: components; totally disconnected
spaces

Compactedness and its applications: the HeineBorel theorem; compactness
of subspaces and product spaces; compactness and sequential compactness

Convergence: the Hausdorff and other separation properties; inadequacy
of sequences; nets; filters and ultrafilters
Texts

S. Lipschutz, "General Topology" (Schaum Outline)
References

A. Armstrong, "Basic Topology" (Springer)

S. Barr, "Experiments in Topology" (Murray)

M. Barnsley, "Fractals Everywhere" (Academic Press)

N. Bourbaki, "General Topology" (Hermann)

V. Bryant, "Metric Spaces, Iteration and Applications" (Cambridge)

E. Copson, "Metric Spaces" (Cambridge)

K. Falconer, "Fractal Geometry" (Wiley)

G. Francis, "A Topological Picturebook" (Springer)

V. Mendelson, "Introduction to Topology" (Alyn & Bacon)

J. Munkres, "Topology, A First Course" (Prentice Hall)

I.M. Singer & J.A. Thorpe, "Lecture Notes on Elementary Topology and
Geometry" (Springer)

W.A. Sutherland, "Introduction to Metric and Topological Spaces" (Oxford)

S. Willard, "General Topology" (Addison Wesley)
MA343 Group Theory (part one)

Group axioms, cyclic groups, permutation groups.

Normal subghroups, homomorphisms, isomorphism theorems.

Permutation groups, isomorphism theorems, linear groups

Direct and semidirect products

Finitely generated Abelian groups

Automorphisms, groups of automorphisms
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