The course aimt at providing the basic notions on topological structures, also useful in the study of Geometry and Analysis.
Expected learning outcomes
Improving of the logical-deductive skills, of the ability to abstract and of the flexibility in facing mathematical problems. Ability to solve specific exercises on the topics presented during the class.
(1) Topological spaces and examples. (2) Basis and associated topology. (3) Metric spaces. (4) Subbases; product topology; subspaces; neighborhoods, closure, limit points (5) Separation axioms (I): T0, T1, T2. Order topology, quotient topology. Topological groups, group actions. (6) Continuous functions; omeomorphisms. (7) Continuity in metric spaces. First countability axiom. (8) Metric spaces: sequences, completeness; completion of a metric space. (9) Compactness. (11) Tychonoff's theorem. (12) Connectedness; path connectedness. (13) Separation axioms (II): T3, T4 . Second countability axiom. Separability. (14) Urysohn's lemma. Completely regular spaces.
Prerequisites for admission
Basic knowledge of Algebra, Linear Algebra and Analysis (suggested courses: Geometria 1, Geometria 2, Analysis 1)
Classroom lessons (36 hours lessons, 20 hours exercises sessions; a tutoring (around ten hours) is also provided).
Lecture notes; textbooks suggested during the class (in particular: J. Munkres, "Topology" (second edition); J. M. Lee, "Introduction to topological manifolds"; Checcucci-Tognoli-Vesentini, "Lezioni di topologia generale").
Assessment methods and Criteria
The final examination consists of two parts, a written exam and an oral exam.
During the written exam, the student must solve some exercises analogous to those presented during the exercises sessions, with the aim of assessing the student's ability to solve problems in topology. The duration of the written exam will be proportional to the number of exercises assigned, also taking into account the nature and complexity of the exercises themselves (however, the duration will not exceed three hours). The oral exam can be taken only if the written component has been successfully passed. In the oral exam, the student will be required to illustrate and demonstrate results presented during the course. Final marks are given using the numerical range 0-30, and will be communicated immediately after the oral examination.