Metric spaces and topology

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Department of Mathematics

Synopsis:

Topology and metric in the plane and Euclidean spaces; convergence, continuous functions and mappings. Metric spaces. Topology of metric spaces, convergence, continuous functions and mappings, Urysohn Lemma, Tietze Theorem. Complete metric spaces, Banach Fixed Point Lemma. Compact metric spaces. Compactness in Euclidean spaces. Lipschitz and Holder functions. Topology on a set. Open and closed sets, closure, boundary. Urysohn Lemma, Tietze Theorem.

Cartesian products, projections. Connected and totally disconnected spaces. Compactness. Tychonoff Theorem for finitely many spaces. Arzelá-Acoli Theorem. Stone-Weierstrass Theorem.

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