GEOMETRY 1
- Overview
- Assessment methods
- Learning objectives
- Contents
- Bibliography
- Teaching methods
- Contacts/Info
It is useful to have followed (and passed the exams of)
Algebra Lineare e Geometria, Algebra 1, Analisi1, Analisi2.
Written and oral examination
Aims of the course and learning skills
Acquisition of the basilar notions of General Topology; n particular the student has to understand the concepts of connection, compactness, and numerability properties of topological spaces. Ability to recognize in concrete cases the topological properties of a space, and the continuity of maps between spaces. Understanding of some basic concepts of Algebraic Topology: in particular omotopy, retraction, and basic techniques for the computation of the fundamental group of a space.
Contents and course program
General Topology:
Topological spaces and their bases. Remindings of metric spaces. Metrizable topologies.
Hausdorff spaces.
Subspace topology.
Internal part, closure, border of a subspace and their properties.
Continuity between topological spaces.
Product topology.
Separation axioms.
Quoteint topology. Group actions on a topological space. Projective spaces.
Connection and arc connection.
Compactness.
Numerability axioms.
Alexandroff compactification.
Elements of Algebraic Topology:
Omotopy of functions and deformation retraction. Omotopical 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.
Teaching methods
Lessons and exercise classes
Textbooks and didactic material
1) M. Manetti, Topologia. Springer, 2008.
2) C. Kosniowski, Introduzione alla topologia algebrica. Zanichelli, 2004 (l'ultima edizione).
Other useful references is are:
Sernesi, Geometria 2
Munkres, Topology (in English).
For exercises and old exams see the personal webpage of the Teacher.
Type of examination
Written and oral exam
Lessons and exercise sessions
To fix an an appointment send me an email.
Borrowed from
click on the activity card to see more information, such as the teacher and descriptive texts.