CODE 25909 2023/2024 8 cfu anno 2 MATEMATICA 8760 (L-35) - GENOVA MAT/03 Italian GENOVA 1° Semester AULAWEB

## 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.

### 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.

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

### Class schedule

The timetable for this course is available here: Portale EasyAcademy

## 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

Data appello Orario Luogo Degree type Note
12/01/2024 10:00 GENOVA Scritto
16/01/2024 10:00 GENOVA Orale
08/02/2024 10:00 GENOVA Scritto
12/02/2024 10:00 GENOVA Orale
14/06/2024 10:00 GENOVA Scritto
18/06/2024 10:00 GENOVA Orale
12/07/2024 10:00 GENOVA Scritto
16/07/2024 10:00 GENOVA Orale
06/09/2024 10:00 GENOVA Scritto
11/09/2024 10:00 GENOVA Orale