SCC0052 - GEOMETRY 1

courses

ID:

SCC0052

Duration (hours):

72

CFU:

SSD:

GEOMETRIA

Year:

2025

Date/time interval

Primo Semestre (22/09/2025 - 16/01/2026)

Course Objectives

The aim of this course is two-fold. On the one hand, one intends to provide the students with a good understanding of the concepts and methods of General Topology, with regard in particular to the study of the main properties of topological spaces such as compactness and connectedness. On the other hand, one intends to introduce the student to the basics of Algebraic Topology, through the study of the concept of homotopy and the notion of fundamental group of a space. EXPECTED LEARNING OUTCOMES: At the end of the course, the student will be able to: 1. understand the concept of continuity in the setting of topological spaces 2. discuss connectedness, compactness and numerability properties of topological spaces 3. recognize in concrete cases the topological properties of a space, and the continuity of maps between spaces. 4. understand some basic concepts of Algebraic Topology: in particular, homotopy, retraction, and basic techniques for the computation of the fundamental group of a space.

Course Prerequisites

It is useful to have attended (and passed the exams of) Algebra Lineare e Geometria, Algebra 1, Analisi 1.

Teaching Methods

Theoretical lectures (frontal) and exercise sessions. Of the six weekly hours, four are devoted to the theoretical lessons, while two are devoted to the discussion of exercises or more specific themes. These exercise sessions are mainly focused on the resolution of problems and on the deepening of particular topics introduced in the theoretical lessons; still, the students have the possibility of asking clarifications or further explanations on the contents of the theoretical lessons. In order to promote an active participation to the exercise sessions, the Lecturer invites the students who wish to do so to present their solutions to the assigned exercises during such sessions.

Assessment Methods

Written and oral examination. The written examination lasts 3 hours and consists of a number of exercises. The written examination is graded on a scale from 0 to 30. Students need a grade of 16/30 or more to be admitted to the oral examination. The oral examination usually begins with the discussion of the written test. The student will then be required to present some of the results seen in class. It will be object of evaluation the ability to present a proof in a complete and rigorous way and to apply these results in concrete cases.

Topological spaces and continuous functions. Bases for a topological space. Subspace topology. Interior, closure, border of a subspace and their properties. Countability and separation axioms. Metric spaces and metrizable topologies. Product topology. Quotients. Group actions on a topological space. Projective spaces. Connectedness and path connectedness. Compactness. Compactifications of topological spaces. Elements of Algebraic Topology: Homotopy of functions and deformation retracts. Homotopical equivalence of topological spaces. Fundamental group of a pointed space. The fundamental group of the circle. A simplified version of the Theorem of Seifert Van Kampen. Application: the fundamental group of the spheres.