Matematica | TOPOLOGY

cod. 0512300036

TOPOLOGY

	0512300036
	DIPARTIMENTO DI MATEMATICA
	EQF6
	MATHEMATICS
	2020/2021

PERCORSO SHARED STUDY PATHWAY

	YEAR OF COURSE 3
	YEAR OF DIDACTIC SYSTEM 2018
	SECONDO SEMESTRE

TEACHING UNITS

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

	Objectives
	THE COURSE "TOPOLOGY" AIMS TO EXPAND SOME FUNDAMENTAL CONCEPTS OF GENERAL TOPOLOGY AND TO INTRODUCE BASIC ALGEBRAIC TOPOLOGY. IT IS KNOWN THAT THE FUNDAMENTAL NOTIONS IN GENERAL TOPOLOGY ARE THA BASIC TOOLS OF A WORKING MATHEMATICIAN IN A VARIETY OF FIELDS. MOREOVER, ALGEBRAIC TOPOLOGY IS ONE OF THE MOST IMPORTANT FIELDS IN MATHEMATICS WHICH USES ALGEBRAIC TOOLS TO STUDY TOPOLOGICAL SPACES. -KNOWLEDGE AND UNDERSTANDING THE FIRST PART IS CONCERNED WITH THE STUDY OF SOME TOPOLOGICAL PROPERTIES TO INTEGRATE AND TO IMPROVE THE ALREADY KNOWN BASIC NOTIONS IN GENERAL TOPOLOGY. THE STUDENTS WILL BE INTRODUCED TO THE LOCAL PROPERTIES, SUCH AS THE LOCAL COMPACTNESS AND THE LOCAL CONNECTNESS, TO THE COMPACTIFICATION PROBLEM, TO THE PARACOMPACT SPACES,.. 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. • TO UNDERSTAND THE FUNDAMENTAL IDEAS IN GENERAL AND ALGEBRAIC TOPOLOGY. • TO EXPLAIN CLEARLY THE FUNDAMENTAL CONCEPTS OF GENERAL AND ALGEBRAIC TOPOLOGY. -APPLYING KNOWLEDGE AND UNDERSTANDING-APPLICATION SKILLS PROBLEMS ARE SUGGESTED AND TYPICAL MATHEMATICAL APPROACHES ARE CONSIDERED TO IMPROVE THE APPLYING ABILITY AND CAPACITY OF INVENTION IN DEMONSTRATION. AT THE END OF THE COURSE THE STUDENTS WILL BE ABLE: • TO ANALYZE PROBLEMS, EXPLAIN CONCEPTS AND MAKE PROOFS. • TO USE EFFICIENTLY THE PROPOSED TECHNIQUES BY THEIR APPLICATION IN CONSTRUCTING SIGNIFICANT EXAMPLES AND IN SOLVING PROBLEMS AND EXERCISES, IN OTHER WORDS BY APPLYING THEM TO PROBLEM-SOLVING.

THE COURSE "TOPOLOGY" AIMS TO EXPAND SOME FUNDAMENTAL CONCEPTS OF GENERAL TOPOLOGY AND TO INTRODUCE BASIC ALGEBRAIC TOPOLOGY.
IT IS KNOWN THAT THE FUNDAMENTAL NOTIONS IN GENERAL TOPOLOGY ARE THA BASIC TOOLS OF A WORKING MATHEMATICIAN IN A VARIETY OF FIELDS. MOREOVER, ALGEBRAIC TOPOLOGY IS ONE OF THE MOST IMPORTANT FIELDS IN MATHEMATICS WHICH USES ALGEBRAIC TOOLS TO STUDY TOPOLOGICAL SPACES.
-KNOWLEDGE AND UNDERSTANDING
THE FIRST PART IS CONCERNED WITH THE STUDY OF SOME TOPOLOGICAL PROPERTIES TO INTEGRATE AND TO IMPROVE THE ALREADY KNOWN BASIC NOTIONS IN GENERAL TOPOLOGY. THE STUDENTS WILL BE INTRODUCED TO THE LOCAL PROPERTIES, SUCH AS THE LOCAL COMPACTNESS AND THE LOCAL CONNECTNESS, TO THE COMPACTIFICATION PROBLEM, TO THE PARACOMPACT SPACES,..
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.
• TO UNDERSTAND THE FUNDAMENTAL IDEAS IN GENERAL AND ALGEBRAIC TOPOLOGY.
• TO EXPLAIN CLEARLY THE FUNDAMENTAL CONCEPTS OF GENERAL AND ALGEBRAIC TOPOLOGY.
-APPLYING KNOWLEDGE AND UNDERSTANDING-APPLICATION SKILLS
PROBLEMS ARE SUGGESTED AND TYPICAL MATHEMATICAL APPROACHES ARE CONSIDERED TO IMPROVE THE APPLYING ABILITY AND CAPACITY OF INVENTION IN DEMONSTRATION. AT THE END OF THE COURSE THE STUDENTS WILL BE ABLE:
• TO ANALYZE PROBLEMS, EXPLAIN CONCEPTS AND MAKE PROOFS.
• TO USE EFFICIENTLY THE PROPOSED TECHNIQUES BY THEIR APPLICATION IN CONSTRUCTING SIGNIFICANT EXAMPLES AND IN SOLVING PROBLEMS AND EXERCISES, IN OTHER WORDS BY APPLYING THEM TO PROBLEM-SOLVING.

	Prerequisites
	PREVIOUS COURSES CONTAINING BASIC CONCEPTS OF MATHEMATICAL ANALYSIS, ALGEBRA AND GEOMETRY ARE PRESUPPOSED. MOREOVER FIRST BASIC NOTIONS IN GENERAL TOPOLOGY. ARE REQUIRED.

	Contents
	GENERAL TOPOLOGY: 1. A BRIEF RECALL OF SOME BASIC TOPOLOGICAL CONCEPTS. 2.LOCAL COMPACT SPACES, LOCAL CONNECTED SPACES. 3.THE COMPACTIFICATION PROBLEM. 4.PARACOMPACT SPACES. 5. METRIZABLE COMPACT SPACES. 6. FUNCTION SPACES. 7. THE EULERO- POINCARE’ CHARACTERISTIC. THE TOPOLOGICAL CLASSIFICATION OF COMPACT, CONNECTED SURFACES (AN INTRODUCTION). ALGEBRAIC TOPOLOGY: 8.FOUNDAMENTAL GROUP AND COVERING SPACES. 9.HOMOTOPY THEORY. 10. THE FOUNDAMENTAL GROUP OF THE CIRCLE. 11. CLASSIFICATION OF TOPOLOGICAL SURFACES IN ALGEBRAIC TOPOLOGY. 12. HOMOLOGY.

Contents

GENERAL TOPOLOGY:
1. A BRIEF RECALL OF SOME BASIC TOPOLOGICAL CONCEPTS.
2.LOCAL COMPACT SPACES, LOCAL CONNECTED SPACES.
3.THE COMPACTIFICATION PROBLEM.
4.PARACOMPACT SPACES.
5. METRIZABLE COMPACT SPACES.
6. FUNCTION SPACES.
7. THE EULERO- POINCARE’ CHARACTERISTIC. THE TOPOLOGICAL CLASSIFICATION OF COMPACT, CONNECTED SURFACES (AN INTRODUCTION).
ALGEBRAIC TOPOLOGY:
8.FOUNDAMENTAL GROUP AND COVERING SPACES.
9.HOMOTOPY THEORY.
10. THE FOUNDAMENTAL GROUP OF THE CIRCLE.
11. CLASSIFICATION OF TOPOLOGICAL SURFACES IN ALGEBRAIC TOPOLOGY.
12. HOMOLOGY.

	Teaching Methods
	TEACHING METHODS ARE BASED ON LESSONS (5CFU) AND DISCUSSIONS (1CFU) ORGANIZED TO SHOW THE ADVANTAGES OF USING TOOLS FROM ABSTRACT ALGEBRA TO STUDY TOPOLOGICAL SPACES. THE BASIC GOAL IS TO FIND ALGEBRAIC INVARIANTS THAT CLASSIFY TOPOLOGICAL SPACES UP TO HOMEOMORPHISM, THOUGH USUALLY MOST CLASSIFY UP TO HOMOTOPY EQUIVALENCE. STUDENTS APPRECIATE HOW FUNDAMENTALS RESULTS, SUCH AS THE FUNDAMENTAL ALGEBRA THEOREM, THE BROWER FIXED POINT THEOREM, ARE EASILY OBTAINED BY USING ALGEBRAIC CONCEPTS THAT ARE TOPOLOGICAL INVARIANTS. THE TEACHING METHODS ARE TARGETED AT INTEGRATING EXPERIENCE, KNOWLEDGE, LEARNING, CURIOSITY AND SKILLS. ASSIGNMENTS, HINTS, SUGGESTIONS ARE GIVEN TO IMPROVE APPLYING ABILITY AND INVENTION IN DEMONSTRATION.

Teaching Methods

TEACHING METHODS ARE BASED ON LESSONS (5CFU) AND DISCUSSIONS (1CFU) ORGANIZED TO SHOW THE ADVANTAGES OF USING TOOLS FROM ABSTRACT ALGEBRA TO STUDY TOPOLOGICAL SPACES. THE BASIC GOAL IS TO FIND ALGEBRAIC INVARIANTS THAT CLASSIFY TOPOLOGICAL SPACES UP TO HOMEOMORPHISM, THOUGH USUALLY MOST CLASSIFY UP TO HOMOTOPY EQUIVALENCE. STUDENTS APPRECIATE HOW FUNDAMENTALS RESULTS, SUCH AS THE FUNDAMENTAL ALGEBRA THEOREM, THE BROWER FIXED POINT THEOREM, ARE EASILY OBTAINED BY USING ALGEBRAIC CONCEPTS THAT ARE TOPOLOGICAL INVARIANTS.
THE TEACHING METHODS ARE TARGETED AT INTEGRATING EXPERIENCE, KNOWLEDGE, LEARNING, CURIOSITY AND SKILLS.
ASSIGNMENTS, HINTS, SUGGESTIONS ARE GIVEN TO IMPROVE APPLYING ABILITY AND INVENTION IN DEMONSTRATION.

	Verification of learning
	A FINAL EXAMINATION AIMS TO VALUE THE KNOWLEDGE OF THE ARGUMENTS TREATED IN THE COURSE, THE LEVEL OF UNDERSTANDING OF PERFORMED MATHEMATICAL APPROACHES, THE COMMUNICATION SKILLS, THE OPENING IN DISCUSSION, THE ORIGINALITY IN ARGUMENTATION AND THE INVENTION IN DEMONSTRATION. IT CONSISTS OF AN ORAL EXAMINATION AIMING TO VALUE NOT ONLY THE ACQUIRED KNOWLEDGES BUT ALSO THE UNDERSTANDING LEVEL AND THE COMMUNICATIONS SKILLS. THE PROFESSOR WILL VERIFY ALL THE GOALS REACHED BY THE STUDENT AND HE WILL EXPRESS THE STUDENT'S LEVEL BY AN OPPORTUNE GRADE. FULL MARKS WILL BE GIVEN TO THE STUDENTS ABLE TO APPLY WITH ORIGINALITY THE ACQUIRED KNOWLEDGES.

Verification of learning

A FINAL EXAMINATION AIMS TO VALUE THE KNOWLEDGE OF THE ARGUMENTS TREATED IN THE COURSE, THE LEVEL OF UNDERSTANDING OF PERFORMED MATHEMATICAL APPROACHES, THE COMMUNICATION SKILLS, THE OPENING IN DISCUSSION, THE ORIGINALITY IN ARGUMENTATION AND THE INVENTION IN DEMONSTRATION.
IT CONSISTS OF AN ORAL EXAMINATION AIMING TO VALUE NOT ONLY THE ACQUIRED KNOWLEDGES BUT ALSO THE UNDERSTANDING LEVEL AND THE COMMUNICATIONS SKILLS.
THE PROFESSOR WILL VERIFY ALL THE GOALS REACHED BY THE STUDENT AND HE WILL EXPRESS THE STUDENT'S LEVEL BY AN OPPORTUNE GRADE.
FULL MARKS WILL BE GIVEN TO THE STUDENTS
ABLE TO APPLY WITH ORIGINALITY THE ACQUIRED KNOWLEDGES.

	Texts
	[1] V.CHECCUCCI, A.TOGNOLI, E.VESENTINI -"LEZIONI DI TOPOLOGIA GENERALE"- FELTRINELLI [2] R.ENGELKING -"GENERAL TOPOLOGY"- HELDERMANN VERLAG 1989 [3] C. KOSNIOWSKI -"INTRODUZIONE ALLA TOPOLOGIA ALGEBRICA" - ZANICHELLI. [4] W.S.MASSEY -" ALGEBRAIC TOPOLOGY: AN INTRODUCTION"- SPRINGER-VERLAG 1991. [5] .MUNKRES -" TOPOLOGY: "-SECOND EDITION PEARSON 2000. [6]S. WILLARD -"GENERAL TOPOLOGY"- ADDISON -WESLEY PUBLISHING COMPANY 1970.

Texts

[1] V.CHECCUCCI, A.TOGNOLI, E.VESENTINI -"LEZIONI DI TOPOLOGIA GENERALE"- FELTRINELLI

[2] R.ENGELKING -"GENERAL TOPOLOGY"- HELDERMANN VERLAG 1989

[3] C. KOSNIOWSKI -"INTRODUZIONE ALLA TOPOLOGIA ALGEBRICA" - ZANICHELLI.

[4] W.S.MASSEY -" ALGEBRAIC TOPOLOGY: AN INTRODUCTION"- SPRINGER-VERLAG 1991.

[5] .MUNKRES -" TOPOLOGY: "-SECOND EDITION PEARSON 2000.

[6]S. WILLARD -"GENERAL TOPOLOGY"- ADDISON -WESLEY PUBLISHING COMPANY 1970.