### Teachings

Select Academic Year:     2017/2018 2018/2019 2019/2020 2020/2021 2021/2022 2022/2023
Professor
GIANLUCA BANDE (Tit.)
Period
First Semester
Teaching style
Convenzionale
Lingua Insegnamento

Informazioni aggiuntive

Course Curriculum CFU Length(h)
[60/64]  MATHEMATICS [64/00 - Ord. 2017]  PERCORSO COMUNE 8 64

### Objectives

KNOWLEDGE AND UNDERSTANDING:
The course aims to teach the basic notions of general topology such as concepts of general topology: metric spaces; topological spaces and topologies; basis of a topology; continuous functions; axioms of numerabilitycountability and separability; product spaces; quotient spaces; compact spaces; connected and path wise connected spaces.

APPLYING KNOWLEDGE AND UNDERSTANDING:
Ability to properly formulate statements about topological spaces and build in a rigorous proof. Knowing how to work with the various topologies studied. Knowing how to properly use the topological properties studied to distinguish topological spaces. Knowing the studied homeomorphisms and knowing how to build some simple homeomorphisms between topological spaces.

JUDGEMENT:
To assimilate the techniques of the general topology to apply them to other contexts such as geometry, differential topology and analysis.

COMMUNICATION SKILLS:
The student will learn to use the language of general topology and rigorously communicate the concepts and the results studied.

LEARNING SKILLS:
Ability to learn to independently solve complex problems and exercises. Ability to read and understand a basic text of general topology. Ability to find the topological properties of (not too complicated) spaces, and know how to use this knowledge to distinguish between two topological spaces, in simple situations.

### Prerequisites

Knowledge of the topology of the real line, real analysis and group theory, linear algebra, analytic geometry, conics and quadrics.

### Contents

Sets and functions between sets; the completeness of the real numbers; the topology of the line; continuous functions from subsets of R to R; definition of metric space; examples of metric spaces; the topology of metric spaces; maps betwen metric spaces; definitons of topology and topological spcaes; examples of topologies and metric spaces; interior, exterior, boundary and closure; definition of a base of a topological space; examples of a basis of a topological space; topology generated by basis: numerability; separation properties; sequences in a topological space; continuous, open and closed maps; homeomorphism and embedding; glueing continuous functions; continuity and sequential continuity; affinity and isometries of the euclidean space; subspaces of the real line and of the plane; convex spaces; topological manifolds; the product topology; prudocts of maps; connected topological spaces; arc connected spaces; connected components; compact topological spaces; compactness in Hausdorff spaces; topological groups and action of a topological group; compactenss in metric spaces; sequentially compact spaces; the quotient topology; quotient spaces; universal property of the quotient topology; examples of quotient space;; topological properties of quotient spaces; the real projective space.

### Teaching Methods

The course includes 64 hours of frontal lectures on the blackboard. The use of surfaces constructed with 3D printers and video projection is expected to stimulate the visualization and intuition of some topological concepts. A lesson register is available on the teaching site and updated after each lesson.

### Verification of learning

Written tests (6 per year) and oral exams. The written test consists in a selection of 3 exercices. Two exercises are similar to those given during the lectures and / or theoretical questions; The third exercise consists in finding the topological properties of a given topological space. Each exercise is worth 10 points and the written test is passed if the final score is not less than 18. In this test mainly occur: knowledge and understanding; knowledge and understanding applied; the ability to learn.
if the score is greater or equal to 18 the student can access to the oral examination.
The oral exam consists of a discussion of at least three arguments. The student must demonstrate that they have understood and assimilated the arguments (knowledge and understanding, knowledge and understanding applied). In addition, students must demonstrate that they are able to explain the concepts and demonstrations learned during the course (Communication Skills). To this end it is necessary that the student, during the oral exam, write on the chalkboard and explain all the steps that it is following to reach the conclusion of an argument (independent judgment). It is advisable to answer exactly the questions without rambling and writing immediately all the necessary details (independent judgment, communication skills). In any case an answer showing strong lacks on the knowledge of course contents or basic notions could compromise the entire oral test.

### Texts

- A. Loi, Introduzione alla topologia generale, Aracne Editrice. (main textbook)
- E. Sernesi – Geometria 2– Bollati Boringhieri
- C. Kosnioski- Introduzione alla Topologia Generale-Zanichelli