Problem Sheets in PDF: Sheet 1 Sheet 2 Sheet 3 Sheet 4 Christmas Sheet Sheet 6 Sheet 7 Sheet 8 Sheet 9 Sheet 10

Course outline:

**Metric spaces.**
Euclidean space in any dimension. Minkowski and Cauchy-Schwarz inequalities.
Metric spaces. Open and closed balls. Continuity of maps between metric spaces.
Convergence of sequences in metric spaces and an equivalent definition of continuity
of maps using sequences. Product metric spaces.
Bounded, open, closed, dense and nowhere dense sets, diameter.
Interior, exterior, boundary and closure of a subset of a metric space.
Another definition of continuity based on open sets.
Complete metric spaces. Cantor's intersection theorem.
Baire's category theorem. Banach's contraction principle.
Metrics on spaces of bounded and continuous functions.
Uniform convergence.
Continuity of a uniform limit of continuous functions.

**Topological spaces.**
Definition. Neighborhoods of points. Limits of sequences.
Hausdorff axiom and uniqueness of limits.
Continuous maps between topological spaces. Homeomorphisms.
Metric topology and metrizable topological spaces.
Induced topology on subspaces.
Stronger and weaker topologies,
discrete and indiscrete topologies as the strongest and the weakest respectively.
Zariski topology on the real line.
Open and closed sets, interior, closure and boundary of a set.
Product topology. Basis in a topological space and abstract basis
defining topology on an abstract set.
Equivalent definitions of continuity of functions
and of limits of sequences using bases.
Local basis. First and second axioms of countability.
Separable metric spaces. Pasting lemma. Quotient topology.

**Compact topological spaces.**
Open coverings, definition.
Compactness in R and R^n, Heine-Borel theorem.
Compactness of closed subsets in compact spaces.
Closedness of compact subsets in Hausdorff spaces.
Continuous functions on compact spaces,
their boundedness and attaining maxima and minima.
Compactness of continuous images of compact spaces.
Compactness of products of compact spaces.
Uniform continuity of continuous functions on compact spaces.
Compactness in the space of continuous functions,
Ascoli-Arzela theorem. Local compactness.
One-point-compactification.

**Connected topological spaces.**
Definitions of connectedness of pathwise connectedness.
Connectedness of unions of connected sets under siutable conditions.
Connectedness of products of connected spaces.
Intermediate value property for continuous functions
on connected spaces.
Connected and path-connected components.
Local connectedness.

**Normed vector spaces.**
Definitions, examples. Banach spaces.
Bounded linear transformations,
equivalent conditions for their continuity.
Convergent series in Banach spaces.
Equivalence of norms. Picard's theorem.

**Elements of homotopy theory.**
Definition of homotopy between two continuous maps.
Homotopy relative to a subset.
Homotopy of paths and loops.
Fundamental group, definition,
group operation. Coverings.
Winding numbers of loops in the circle
and and of loops in the plain with a point removed.

For exam-related problems look in TCD past examination papers and Mathematics department examination papers.