Luca VITAGLIANO | TOPOLOGIA

cod. 0512300036

TOPOLOGIA

	0512300036
	DIPARTIMENTO DI MATEMATICA
	EQF6
	MATHEMATICS
	2016/2017

CURRICULUM MATEMATICA AD INDIRIZZO APPLICATIVO Cohort 2014/2015

	YEAR OF COURSE 3
	YEAR OF DIDACTIC SYSTEM 2010
	SECONDO SEMESTRE

TEACHING UNITS

	SSD	CFU	HOURS	ACTIVITY	TYPE OF ACTIVITY
	MAT/03	6	48	LESSONS	COMPULSORY SUBJECTS, CHARACTERISTIC OF THE CLASS

PROFESSORS

	ANNAMARIA MIRANDA T Curriculum
	LUCA VITAGLIANO Curriculum

	Objectives
	THE COURSE "TOPOLOGY" AIMS TO INTRODUCE STUDENTS TO THE FUNDAMENTAL CONCEPTS OF GENERAL TOPOLOGY AND ALGEBRAIC TOPOLOGY. -KNOWLEDGE AND UNDERSTANDING THE FIRST PART IS CONCERNED WITH THE STUDY OF TOPOLOGICAL SPACES AND THEIR STRUCTURE-PRESERVING FUNCTIONS. THE STUDENTS WILL BE INTRODUCED TO THE GENERAL STRUCTURE OF A TOPOLOGICAL SPACE, THE CONSTRUCTION OF NEW TOPOLOGICAL SPACES FROM OLD, THE TOPOLOGICAL PROPERTIES INVARIANTS UNDER CONTINUOUS MAPPINGS. THE BASIC GOAL OF THE SECOND PART IS TO STUDY SOME PROPERTIES OF TOPOLOGICAL SPACES AND MAPS BETWEEN THEM BY ASSOCIATING ALGEBRAIC INVARIANTS TO EACH SPACE. TWO WAYS IN WHICH THIS CAN BE DONE ARE THROUGH FUNDAMENTAL GROUPS, OR MORE GENERALLY HOMOTOPY THEORY, AND THROUGH HOMOLOGY AND COHOMOLOGY GROUPS. ON SATISFYING THE REQUIREMENTS OF THIS COURSE, THE STUDENT WILL HAVE THE FOLLOWING KNOWLEDGE AND SKILLS. • UNDERSTAND THE FUNDAMENTAL IDEAS IN GENERAL AND ALGEBRAIC TOPOLOGY. •EXPLAIN CLEARLY THE FUNDAMENTAL CONCEPTS OF GENERAL AND ALGEBRAIC TOPOLOGY. -APPLICATION SKILLS THE STUDENT WILL BE ABLE TO: •SHOW AN EFFICIENT USE OF TOPOLOGY TECHNIQUES, BY APPLYING THEM TO PROBLEM-SOLVING. •DEMONSTRATE CAPACITY FOR MATHEMATICAL REASONING THROUGH ANALYZING, PROVING AND EXPLAINING PROPOSITIONS AND CONCEPTS FROM TOPOLOGY.

THE COURSE "TOPOLOGY" AIMS TO INTRODUCE STUDENTS TO THE FUNDAMENTAL CONCEPTS OF GENERAL TOPOLOGY AND ALGEBRAIC TOPOLOGY.

-KNOWLEDGE AND UNDERSTANDING
THE FIRST PART IS CONCERNED WITH THE STUDY OF TOPOLOGICAL SPACES AND THEIR STRUCTURE-PRESERVING FUNCTIONS. THE STUDENTS WILL BE INTRODUCED TO THE GENERAL STRUCTURE OF A TOPOLOGICAL SPACE, THE CONSTRUCTION OF NEW TOPOLOGICAL SPACES FROM OLD, THE TOPOLOGICAL PROPERTIES INVARIANTS UNDER CONTINUOUS MAPPINGS.
THE BASIC GOAL OF THE SECOND PART IS TO STUDY SOME PROPERTIES OF TOPOLOGICAL SPACES AND MAPS BETWEEN THEM BY ASSOCIATING ALGEBRAIC INVARIANTS TO EACH SPACE. TWO WAYS IN WHICH THIS CAN BE DONE ARE THROUGH FUNDAMENTAL GROUPS, OR MORE GENERALLY HOMOTOPY THEORY, AND THROUGH HOMOLOGY AND COHOMOLOGY GROUPS.
ON SATISFYING THE REQUIREMENTS OF THIS COURSE, THE STUDENT WILL HAVE THE FOLLOWING KNOWLEDGE AND SKILLS.
• UNDERSTAND THE FUNDAMENTAL IDEAS IN GENERAL AND ALGEBRAIC TOPOLOGY.
•EXPLAIN CLEARLY THE FUNDAMENTAL CONCEPTS OF GENERAL AND ALGEBRAIC TOPOLOGY.

-APPLICATION SKILLS
THE STUDENT WILL BE ABLE TO:
•SHOW AN EFFICIENT USE OF TOPOLOGY TECHNIQUES, BY APPLYING THEM TO PROBLEM-SOLVING.
•DEMONSTRATE CAPACITY FOR MATHEMATICAL REASONING THROUGH ANALYZING, PROVING AND EXPLAINING PROPOSITIONS AND CONCEPTS FROM TOPOLOGY.

	Prerequisites
	PREVIOUS COURSES CONTAINING BASIC CONCEPTS OF MATHEMATICAL ANALYSIS AND ALGEBRA ARE PRESUPPOSED.

	Contents
	GENERAL TOPOLOGY: 1.TOPOLOGICAL SPACES. 2.CONTINUITY AND HOMEOMORPHISMS, TOPOLOGICAL INVARIANCE. 3.CONSTRUCTION OF TOPOLOGICAL SPACES FROM DIFFERENT POINTS OF VIEW (AMONG OTHERS PRODUCT TOPOLOGY, QUOTIENT TOPOLOGY, TOPOLOGY DETERMINED BY A BASIS OR SUBBASIS). 4.SEPARATION PROPERTIES, COMPACTNESS, CONNECTEDNESS. ALGEBRAIC TOPOLOGY: 5.FOUNDAMENTAL GROUP AND COVERING SPACES. 6.HOMOTOPY THEORY. 7. THE FOUNDAMENTAL GROUP OF THE CIRCLE. 8.CLASSIFICATION OF TOPOLOGICAL SURFACES.

Contents

GENERAL TOPOLOGY:
1.TOPOLOGICAL SPACES.
2.CONTINUITY AND HOMEOMORPHISMS, TOPOLOGICAL INVARIANCE.
3.CONSTRUCTION OF TOPOLOGICAL SPACES FROM DIFFERENT POINTS OF VIEW (AMONG OTHERS PRODUCT TOPOLOGY, QUOTIENT TOPOLOGY, TOPOLOGY DETERMINED BY A BASIS OR SUBBASIS).
4.SEPARATION PROPERTIES, COMPACTNESS, CONNECTEDNESS.
ALGEBRAIC TOPOLOGY:
5.FOUNDAMENTAL GROUP AND COVERING SPACES.
6.HOMOTOPY THEORY.
7. THE FOUNDAMENTAL GROUP OF THE CIRCLE.
8.CLASSIFICATION OF TOPOLOGICAL SURFACES.

	Teaching Methods
	•LESSONS. •EXERCISES.

	Verification of learning
	THE PROFESSOR WILL VERIFY, BY AN ORAL EXAMINATION, ALL THE GOALS REACHED BY THE STUDENT. HE WILL EXPRESS STUDENT'S KNOWLEDGE AND COMPETENCE BY AN OPPORTUNE GRADE.

	Texts
	1.M.A. ARMSTRONG BASIC TOPOLOGY UNDERGRADUATE TEXTS IN MATHEMATICS SPRINGER-VERLAG 1983. 2. R. ENGELKING GENERAL TOPOLOGY HELDERMANN VERLAG 1989. 3. W.S.MASSEY ALGEBRAIC TOPOLOGY: AN INTRODUCTION SPRINGER-VERLAG 1991. 4.S. WILLARD GENERAL TOPOLOGY ADDISON -WESLEY PUBLISHING COMPANY 1970.

	More Information
	•E-MAIL ADDRESS OF THE PROFESSOR: AMIRANDA@UNISA.IT •WEBSITE ADDRESS OF THE PROFESSOR: HTTP://WWW.UNISA.IT/DOCENTI/ANNAMARIAMIRANDA/INDEX

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Luca VITAGLIANO | TOPOLOGIA