OVERVIEW

The course offers an introduction to General Topology.

AIMS AND CONTENT

LEARNING OUTCOMES

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

AIMS AND LEARNING OUTCOMES

At the end of the course, students will have a good understanding of the fundamental notions of General Topology, such as continuity, separation axioms, connectedness, compactness, metrizability.

PREREQUISITES

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

TEACHING METHODS

Traditional method: lectures in presence.

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

2. M. Manetti, Topologia, seconda edizione, Springer, 2014;

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

TEACHERS AND EXAM BOARD

Exam Board

STEFANO VIGNI (President)

FABIO TANTURRI

LESSONS

EXAMS

EXAM DESCRIPTION

Written and oral exam.

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