Course information

Real numbers: Natural numbers, Peano axioms, Integers, Rationals, Real numbers, properties, Dedekind cuts. Metric spaces: Definition, examples, metrics in vector spaces defined by norms. Sequences and functions: Convergence of sequences, Continuous functions. Open and closed subsets of metric spaces: Accumulation points of a set, Open and closed subsets, Characterizations of continuity, Equivalent metrics. Dense subsets and separable metric spaces: Countable and uncountable sets, Zorn s Lemma, Dense subsets and separable metric spaces, Bases of neighborhoods. Complete metric spaces: Completeness, Baire s theorem, Uniformly continuous functions. Compact metric spaces: Properties of compact spaces, continuous functions in compact metric spaces, connectedness. Totally bounded subsets of metric spaces. Sequences of functions: Pointwise convergence, uniform convergence of a sequence of real functions. The spaces C[a,b]: Normed vector spaces, The vector space C[a,b], Equicontinuous families of functions, Arzela s theorem. Products of Metric spaces: Finite and infinite countable products of metric spaces, the Cantor set.