(1) Topological spaces and examples.
(2) Basis ,properties and generated topology. Comparison of topologies. Euclidean topology on the real line. Other topologies on R
(3) Metric spaces, distances, examples. Induced topology. Uniform convergence topology
(4) Subbasis ant the topology they generate Pruduct topology. Examples of equivalent topologies. In R _ R Canonical projections. Subspaces, basis ,properties. Interior, closure limit points clopen. Closure operators and the topology they generate. Isiolated points, perfect sets dense sets boundary Weierstrass approximation theorem( )
Separation axioms: T0, T1, T2 . Properties, examples, the case of metric spaces. Properties of T1 and T2 spaces.
(5) Order topology , quotient topology. Topological groups and their actions Fundamental systems of neighborhouds their properties , generated topology. (_).
.(6) Continuous functions, compositons omeomorphisms, quotients and their universal property. Examples. Locally finite famiglie. Products and their projections.
(7) Topological vector spaces, metric induced by a norm ( ). Continuity in metric spaces. Sequences and limits. First numerabilità axiom and related results.
(8) Metric spaces , sequenze and Cauchy sequenze. Completeness, Cantor completeness.( ). Isometries
The functional space Y I and its uniform metric corresponding to the metric d on Y. The sup matric completeness of B(X;R). Completeness of a metric space (X,d) by by isometrically immerging it into(B(X;R);
(9) Compactness. First properties, coverings, subspaces closed sets in compact spaces. Compactness and continuous functions. A peano curve( ).
(10) Product of compact spaces examples . compactness and the order topology. The case of Rn. Uncountability of compact Hausdorff spaces made only of limit points.( ).Frechet compactness and its relation with sequential compactness. Equivalent notions of compactness in metric spaces. Totally bounded spaces, Lebesgue number of an open covering. Equicontinuous functions, Ascoli Theorem.
. Algebras and lattices. Stone extension of Weirstrass Theorem. (_).
(11) Tychonoff tepore and its consequences.. Box topology vs. product topology. Compact-open topology and local compactness.( ). Properties.. Compact-open topology and puntual convergence topology. Generalised Ascoli Theorem.
(12)Separation connected spaces ,examples ,properties. Unions ,intersections, closure images under a continuous map.Products in the product topology and in the box topology. Mean value theorem, paths.
The pasting lemma.
(13) Separation axioms : T3 ,T4. Characterizations. Metric spaces subspaces, Second numerability axiom. Lindeloff Theorem. Separable spaces, well ordered spaces.
(14) Urysohn Lemma. T4 and separability by continuous functions. Completely regular spaces. Urysohn metrisation Theorem. Characterisation of completely regular spaces. Tietze extension Theorem.
(15) Compactifications, Alexandroff and Stone -Cech compactifications (_) .
(16) Omotopies, equivalence relation, paths. Product of paths.Fundamental group. The fundamental group. Simply connected spaces Universal covering. Covering maps. Some algebra: free Abelian groups,, free products. Seifert-Van Kampen Theorem( )