# GEOMETRY 2A

## Learning outcomes of the course unit

The main goal of the course is the study of the main properties of topological spaces and a first introduction to Algebraic Topology.

At the end of the course, students will be familiar with the basics notions of connection, compactness, separation axioms, and numerability, with the product topology and the quotient topology, with metric spaces, with the theory of homotopy, of the fundamental group and of the coverings. They will also be able to deal with the resolution of problems of a theoretical and practical nature in the field of General Topology and of Algebraic Topology.

## Prerequisites

Analysis 1, Geometry 1

## Course contents summary

Geometria 2, first part: General Topology, Algebraic Topology.

## Course contents

1. Topogical spaces.

1.1. Topologies on a set. Open sets and closed sets.

1.2 Neighbourhoods. Fundamental systems of Neighbourhoods.

1.3. Trivial topologies. Euclidean topology, cofinite topology, topology of the lower and upper semicontinuity.

1.4 Points of accumulation. Closure, Derivative, Internal and border of a set. Sequences in a topological space. Convergent sequences.

1.5. Basis of a topological space. Sub-bases. Topology generated by a class of subsets.

1.6. Topological subspaces.

1.7 Distances on a set.

1.8 Metric spaces.

2. Continuos Maps.

2.1. Continuous maps between topological spaces.

2.2. Continuity at a point.

2.3. Open, closed maps. Homeomorphisms.

3. Product topology.

3.1. Product of a finite number of topological spaces. Basis for the topological product.

3.2. Product of an arbitrary family of topological spaces.

3.2. Product of a countable infinity of metric spaces.

4. Quotient topology.

4.1. Quotient of topological spaces. Canonical projection.

4.2. Saturated. Correspondence between open saturated and open quotient topology.

4.3. Real projective space. Complex projective space. The Grassmannians.

4.4. The n-dimensional torus.

5. The axioms of separation and numerability.

5.1. The axiom T0 and the axiom T1.

5.2. The axiom of Hausdorff.

5.3. Regular spaces and normal spaces.

5.4 1-countable spaces.

5.5. 2-countable spaces. Lindelöff's theorem.

6. Connection.

6.1. Topological spaces connected.

6.2. The connected ones of the real line.

6.3. Connection and continuous maps.

6.4. Families of connected subsets. Product of connected spaces.

6.5. Locally connected spaces. The grid.

6.6. Path-connected spaces.

7. Compactness.

7.1. Open covering of a topological space. Almost compact spaces and compact spaces.

7.2. Continuous maps and compactness.

7.3. The Heine-Borel Theorem.

7.4. The Tychonoff Theorem.

7.5 The compact sets of the Euclidean real space.

7.6 The Bolzano-Weierstrass Theorem.

7.7. Compact metric spaces.

7.8. Alexandroff compactification of a topological space.

8. Metric spaces.

8.1. Complete metric spaces.

8.2. Completion of a metric space.

8.3. Extension of continuous maps.

8.4 The Contraction Theorem.

8.5. The Lemma of Urysohn.

8.6. The Lemma of Baire.

9. Homotopy, fundamental group and coating spaces.

9.1. The lemma of gluing.

9.2. Homotopy between maps.

9.3. Homotopy between paths.

9.4. The fundamental group of a topological space.

9.5. Functorial properties of the fundamental group.

9.6. The fundamental group of the circumference.

9.7. Coverings of topological spaces.

9.8. G-spaces. Fundamental group of a G-space.

9.9. The Theorem by Seifert-Van Kampen.

## Recommended readings

V. Checcucci, A. Tognoli, E. Vesentini, Lezioni di topologia generale, Feltrinelli, 1976.

E. Sernesi, Geometria 2, Bollati Boringhieri, 1998.

J.R. Munkres, Topology, Pearson College Div; 2 edizione, 2017.

J.R. Munkres, Elements of Algebraic Topology, Francis and Taylor, 1996.

C. Kosniowski, Introduzione alla topologia algebrica, Zanichelli, 1988.

## Teaching methods

Lectures and exercises. The activities will take place in telepresence through the use of the Teams and Elly platforms. In particular, they will be

performed lessons in synchronous mode (via Teams). During

lessons in synchronous mode (direct), mainly frontal moments will alternate with interactive moments with the

students. To promote active participation in the course, various activities will be proposed both individual and a

small group, through the use of the resources present in Elly. During the lectures, the

topics will be presented in a formal and rigorous way. The course will give particular emphasis to the application and calculation aspects, while not neglecting

the theoretical aspect. For this purpose, the exercises carried out in the classroom in which the student learns to apply the

theory seen in class when solving concrete problems.

## Assessment methods and criteria

The final exam, including the first and second part of the Geometry 2 course, consists of a written test and an oral exam. In place of the written exam, students can take two intermediate tests. The evaluation of the intermediate tests and the written test is as follows: students who score between 24 and 30, achieve A.

Students who score between 18 and 23, achieve B. Students who score below 18, achieve C. The written test is considered passed when at least B. Students who achieve at least B in the two intermediate tests have access directly to the oral exam, which can be carried out in any call of the academic year of reference.

The oral exam consists in the proofs of significant theorems and / or in the exposition of topics, definitions, treated in the lectures.

## Other informations

Exercises and problems to be performed outside of class hours will be assigned by the teacher.

Notes from the teacher will be distributed. The course notes in PDF format and all the material used during the lessons and exercises are made available to the students on the Elly educational platform.