# General Topology

Academic Year 2023/2024 - Teacher: Angelo BELLA

## Expected Learning Outcomes

Training in the use of formal language in abstract mathematics. The course provides a complete description of the basic facts of General Topology. Particular emphasis will be given to the discussion of examples and exercises.

At the conclusion of the course, the students shuold be able to understand the basic notions, to apply their knowledge and understanding. They also should be able to give oral and written presentation of the most important theorems of the contents of the course as well as to work both in collaboration with other people and by themselves, making judgements.

Learning assessment may also be carried out on line, should the conditions require it.

## Course Structure

Lectures with slides and exercises in which the assigned exercises are corrected.

If the teaching is given in a mixed or remote way, the necessary changes may be introduced with respect to what was previously stated, in order to comply with the program envisaged and reported in the syllabus.

PLEASE NOTE: Information for students with disabilities and / or SLD

To guarantee equal opportunities and in compliance with the laws in force, interested students can ask for a personal interview in order to plan any compensatory and / or dispensatory measures, based on the didactic objectives and specific needs.

It is also possible to contact the referent teacher CInAP (Center for Active and Participatory Integration - Services for Disabilities and / or SLD) of our Department, prof. Filippo Stanco

## Required Prerequisites

basic logic properties

## Attendance of Lessons

strongly recommended

## Detailed Course Content

The notion of topological space. Open and closed sets. Bases and fundamental systems of neighborhoods. Construction of a topology. First and second axioms of countability. Continuous functions and homeoformisms. Subspaces and hereditary properties. Product of topological spaces: the finite case and the general case. Quotient spaces. Metric spaces and metrizable spaces. Separation axioms. Normal spaces and Urysohn's lemma. The Tietze extension theorem. Compact spaces and their fundamental properties. Tychonoff's theorem. The embedding theorem. A fundamental characterization of complete regularity. The notion of compactification. Connected spaces and their properties. Connectidness of a product. Locally compact spaces and by Aleksandroff's compactification.

## Textbook Information

1. Professor's notes.

2.Topologia by M. Manetti. General Topology by R. Engelking.

## Course Planning

SubjectsText References
1La nozione di spazio topologico. 1
2Insiemi aperti e chiusi. Basi e sistemi fondamentali di intorni. 1
3Costruzione di una topologia. Primo e secondo assioma di numerabilità. 1
4. Funzioni continue ed omeoformismi. Sottospazi e proprietà ereditarie.1
5Prodotto di spazi topologici: il caso finito e il caso generale. Spazi quoziente. Spazi metrici e spazi metrizzabili.1
6Assiomi di separazione. Spazi normali e lemma di Urysohn. Il teorema di estensione di Tietze1
7Spazi compatti e loro proprietà fondamentali. Il teorema di Tychonoff. Il teorema di immersione1
8Una caratterizzazione fondamentale della completa regolarità. La nozione di compattificazione. Spazi connessi e loro proprietà. 1
9La connessione di un prodotto. Spazi loalmente compatti e compattificazione di Aleksandroff.1

## Learning Assessment

### Learning Assessment Procedures

oral exam

Learning assessment may also be carried out on line, should the conditions require it.