Acquire a good knowledge of concepts and methods of general topology, with particular regard to the study of the main properties of topological spaces such as connection and compactness. Introduce the student to the basic elements of algebraic topology, through the introduction of the fundamental group and the topological classification of curves and surfaces.
Curriculum
teacher profile teaching materials
Topological spaces.
Bases for a topology.
Euclidean topology. Cofinite topology.
Finite and infinite products of topological spaces.
Connected spaces.
Connectedness of intervals.
Path-connected spaces.
Compact spaces.
Heine-Borel theorem.
Metric spaces.
Complete spaces. Totally bounded spaces.
Compactness in metric spaces.
Separation properties in metric spaces.
Metrizable spaces.
Homotopy of functions.
Contractible spaces.
Paths and path equivalence.
Fundamental group.
Functorial properties of the fundamental group.
Dependence of the fundamental group on the base point.
Covering spaces.
Lifts.
Fundamental group of the circle.
Fundamental group of a product.
Homotopy between topological spaces.
Seifert-van Kampen theorem for simply connected spaces.
Fundamental group of spheres.
Homotopy between spheres and Euclidean spaces with a point removed.
Supplementary book: Topology James R. Munkres - Prentice Hall.
Programme
Topological spaces.
Bases for a topology.
Euclidean topology. Cofinite topology.
Finite and infinite products of topological spaces.
Connected spaces.
Connectedness of intervals.
Path-connected spaces.
Compact spaces.
Heine-Borel theorem.
Metric spaces.
Complete spaces. Totally bounded spaces.
Compactness in metric spaces.
Separation properties in metric spaces.
Metrizable spaces.
Homotopy of functions.
Contractible spaces.
Paths and path equivalence.
Fundamental group.
Functorial properties of the fundamental group.
Dependence of the fundamental group on the base point.
Covering spaces.
Lifts.
Fundamental group of the circle.
Fundamental group of a product.
Homotopy between topological spaces.
Seifert-van Kampen theorem for simply connected spaces.
Fundamental group of spheres.
Homotopy between spheres and Euclidean spaces with a point removed.
Core Documentation
Text: Lezioni di topologia Lucia Caporaso - Available on TeamsSupplementary book: Topology James R. Munkres - Prentice Hall.
Attendance
In classroom, with 3/4 meetings per week.Type of evaluation
Written exam with exercises and oral exam on the theory. teacher profile teaching materials
Programme
see Prof Caporaso's pageCore Documentation
see Prof Caporaso's pageAttendance
see Prof Caporaso's pageType of evaluation
see Prof Caporaso's page teacher profile teaching materials
Topological spaces.
Bases for a topology.
Euclidean topology. Cofinite topology.
Finite and infinite products of topological spaces.
Connected spaces.
Connectedness of intervals.
Path-connected spaces.
Compact spaces.
Heine-Borel theorem.
Metric spaces.
Complete spaces. Totally bounded spaces.
Compactness in metric spaces.
Separation properties in metric spaces.
Metrizable spaces.
Homotopy of functions.
Contractible spaces.
Paths and path equivalence.
Fundamental group.
Functorial properties of the fundamental group.
Dependence of the fundamental group on the base point.
Covering spaces.
Lifts.
Fundamental group of the circle.
Fundamental group of a product.
Homotopy between topological spaces.
Seifert-van Kampen theorem for simply connected spaces.
Fundamental group of spheres.
Homotopy between spheres and Euclidean spaces with a point removed.
Supplementary book: Topology James R. Munkres - Prentice Hall.
Programme
Topological spaces.
Bases for a topology.
Euclidean topology. Cofinite topology.
Finite and infinite products of topological spaces.
Connected spaces.
Connectedness of intervals.
Path-connected spaces.
Compact spaces.
Heine-Borel theorem.
Metric spaces.
Complete spaces. Totally bounded spaces.
Compactness in metric spaces.
Separation properties in metric spaces.
Metrizable spaces.
Homotopy of functions.
Contractible spaces.
Paths and path equivalence.
Fundamental group.
Functorial properties of the fundamental group.
Dependence of the fundamental group on the base point.
Covering spaces.
Lifts.
Fundamental group of the circle.
Fundamental group of a product.
Homotopy between topological spaces.
Seifert-van Kampen theorem for simply connected spaces.
Fundamental group of spheres.
Homotopy between spheres and Euclidean spaces with a point removed.
Core Documentation
Text: Lezioni di topologia Lucia Caporaso - Available on TeamsSupplementary book: Topology James R. Munkres - Prentice Hall.
Attendance
In classroom, with 3/4 meetings per week.Type of evaluation
Written exam with exercises and oral exam on the theory. teacher profile teaching materials
Programme
see Prof Caporaso's pageCore Documentation
see Prof Caporaso's pageAttendance
see Prof Caporaso's pageType of evaluation
see Prof Caporaso's page