OVERVIEW

The cours offers an introduction to general topology. In particular, it teaches definitions and properties of metric spaces, topological spaces and continuous functions between them. It is a course for second year students, the notions and competences developed in this course will be helpful for subsequent courses.

AIMS AND CONTENT

LEARNING OUTCOMES

The course aims to introduce the student to the foundations of the General Topology, with particular attention to the concepts of continuity, connectivity and compactness.

AIMS AND LEARNING OUTCOMES

The goal of this course is to intruce to the techniques and to the study of general topology. In particolare the goals are:

• to introduce to the theory of topological spaces
• to study the main homeomorphism invariants of topological spaces
• to introduce to the theory of metrizability
• to give several examples of topological spaces.

Motivations and historical backgrounds on the beginning of topology will be partially aimed.

Moreover, the course aims to train:

• the ability to use in a precise way the language of topology
• the ability to formalize geometrical problems in terms of topology
• the ability to prove simple theorems in topology.

At the end of the couse the student will be able to:

• calculate the main invariants of a topological space
• calculate the quotient of a topological space under the action of a group
• establish if a given topological space has properties like compactness, connectedness, path-connectedness, separation axioms

PREREQUISITES

The courses of the first year of our Laurea in Matematica.

TEACHING METHODS

The goal of the lectures is to present the theoretical part of the course, as well as providing solutions to problems, whose aim is to help explain better the theory. Attendance at lectures and exercises is strongly recommended.
Working students and students with certified specific learning disorders (SLD), disabilities, or other special educational needs are encouraged to contact the instructor at the beginning of the course to agree on teaching and assessment methods that, while respecting the learning objectives, take into account individual learning styles

SYLLABUS/CONTENT

• Metric spaces: first properties.
• Continuous maps between metric spaces; isometries.
• Topological spaces: first properties.
• Interior and closure of a subset of a topological space.
• Bases of open sets and fundamental systems of neighbourhoods.
• Axioms of countability.
• Sequences in topological spaces.
• Continuous maps between topological spaces; homeomorphisms.
• Subspaces of a topological space.
• (Arbitrary) productss of tpological spaces.
• Quotients of topological spaces.
• Separation axioms (in particular: Hausdorff spaces).
• Connectedness; local connectedness.
• Compactness; local compactness.
• Tychonoff's theorem (for arbitrary products).
• Countable compactness; sequential compactness.
• Alexandroff compactification.
• Equivalence for metrizable spaces of the notions of compactness, countable compactness and sequential compactness.
• Complete metric spaces.
• Completion of a metric space.
• Urysohn's lemma.
• Urysohn's metrizability theorem.
• Tietze's theorem.
• Baire spaces.

RECOMMENDED READING/BIBLIOGRAPHY

1. M. Manetti, Topologia, seconda edizione, Springer, 2014.

2.  C Kosniowski: Introduzione alla topologia algebrica , Zanichelli.

3. S. Willard, General topology, Dover, 2004. 

4. V. Checcucci, A. Tognoli, A. Vesentini, Lezioni di topologia generale, Feltrinelli, 1968.

5. 3. W.S. Messey: A basic Course in Algebraic Topology , Springer.

TEACHERS AND EXAM BOARD

LESSONS

LESSONS START

22-set-2025

Class schedule

The timetable for this course is available here: EasyAcademy

EXAMS

EXAM DESCRIPTION

Written and oral exam.

Students with DSA certification ("specific learning disabilities"), disability or other special equcational needs are advised to contact the professor at the beginning of the course and agree on the teaching and examination methods that are in compliance with the main teaching objectives and takes into account individual learning arrangements and provides appropriate compensatory tools.

ASSESSMENT METHODS

The written part of the exam will consist in exercises on the contents of the course.

The oral part of the exam will be based on the contents of the course and will assess the overall knowledge of the student.

Exam schedule

FURTHER INFORMATION

For any additional information not included in the course description, please contact the instructor