Training in the use of formal language in abstract mathematics. The course provides a complete description of the basic facts of General Topology. Particular emphasis will be given to the discussion of examples and exercises.
Lectures with slides and exercises in which the assigned exercises are corrected.
If the teaching is given in a mixed or remote way, the necessary changes may be introduced with respect to what was previously stated, in order to comply with the program envisaged and reported in the syllabus.
The notion of topological space. Open and closed sets. Bases and fundamental systems of neighborhoods. Construction of a topology. First and second axioms of countability. Continuous functions and homeoformisms. Subspaces and hereditary properties. Product of topological spaces: the finite case and the general case. Quotient spaces. Metric spaces and metrizable spaces. Separation axioms. Normal spaces and Urysohn's lemma. The Tietze extension theorem. Compact spaces and their fundamental properties. Tychonoff's theorem. The embedding theorem. A fundamental characterization of complete regularity. The notion of compactification. Connected spaces and their properties. Connectidness of a product. Locally compact spaces and by Aleksandroff's compactification.
1. Professor's notes.
2.Topologia by M. Manetti. General Topology by R. Engelking.