Karta przedmiotu

Topology of Metric Spaces

Some basic information about the module

Cycle of education:

2019/2020

The name of the faculty organization unit:

The faculty Mathematics and Applied Physics

The name of the field of study:

Mathematics

The area of study:

sciences

The profile of studing:

The level of study:

first degree study

Type of study:

full time

discipline specialities :

Applications of Mathematics in Economics

The degree after graduating from university:

bachelor's degree

The name of the module department :

Departament of Topology and Algebra

The code of the module:

4081

The module status:

mandatory for teaching programme Applications of Mathematics in Economics

The position in the studies teaching programme:

sem: 3 / W30 C30 / 5 ECTS / E

The language of the lecture:

Polish

The name of the coordinator:

Prof. Dov Bronisław Wajnryb, DSc, PhD

office hours of the coordinator:

wtorek 10:30 - 12 czwartek 10:30 - 12

semester 3:

Janusz Dronka, PhD , office hours wed. 12:15 - 13:45 room L - 108 e thur. 10:30 - 12 room L - 108 e

The aim of studying and bibliography

The main aim of study:
To teach basic notions of topology of metric spaces and their properties. Metric, open and closed sets, sequences, complete spaces, connected spaces, compact spaces. Continuous functions and their properties.

The general information about the module:
regular studies, semester III, lectures 30 hours, exercises 30 hours, ends with an exam.

Bibliography required to complete the module

Bibliography used during lectures
1	K. Kuratowski,	Wstęp do teorii mnogości i topologii,	PWN, Warszawa .	2004

Bibliography used during classes/laboratories/others
1	. R. Engelking, K. Sieklucki	Wstęp do topologii	PWN, Warszawa.	1986

Bibliography to self-study
1	. R. Duda, , PWN, Warszawa 1986.	Wprowadzenie do topologii. Część I: Topologia ogólna	PWN, Warszawa .	1986

Basic requirements in category knowledge/skills/social competences

Formal requirements:
The student satisfies the formal requirements set out in the study regulations

Basic requirements in category knowledge:
countable and uncountable sets, operations on sets, implication and its negation, functions and relations, limit of a sequence, the limit and continuity of a function of one and two variables.

Basic requirements in category skills:
Student can compute the limit of a simple sequence, can verify whether the sequence tends to infinity, can compute a limit of a function, can find the sum and the intersection of two sets.

Basic requirements in category social competences:
Has the ability to define priorities needed for the realization of a particular task, determined by himself/herself or by others.

Module outcomes

MEK	The student who completed the module	Types of classes / teaching methods leading to achieving a given outcome of teaching	Methods of verifying every mentioned outcome of teaching	Relationships with KEK	Relationships with PRK
MEK01	Knows the basic notions of topology: metric space, open and closed sets, dense set, the boundary and the interior of a set, a limit point of a set.	lectures and exercises	Questions at the blackboard during exercises. Problems to solve at test, also theoretical: definitions, formulations of theorems. Similar practical and theoretical questions during exam.	K-W01+ K-W06+ K-U23+ K-K01+	P6S-KK P6S-UW P6S-WG P6S-WK
MEK02	Knows the notions of complete space, compact space, connected space, separable space and their properties.	lectures and exercises	Questions at the blackboard during exercises. Problems to solve at test, also theoretical: definitions, formulations of theorems. Similar practical and theoretical questions during exam.	K-U23+ K-K01+	P6S-KK P6S-UW
MEK03	Knows the notion of a continuous function, knows the basic properties of the continuous functions, knows the theorems on continuous functions on a compact space.	lectures and exercises	Questions at the blackboard during exercises. Problems to solve at test, also theoretical: definitions, formulations of theorems. Similar practical and theoretical questions during exam.	K-W03+ K-W04++ K-U23+ K-K01+	P6S-KK P6S-UW P6S-WG P6S-WK
MEK04	Can prove simple properties and theorems, e.g. a convergent sequence is bounded, sum of two closed sets is closed, a compact subset is closed.	lectures and exercises	Questions at the blackboard during exercises. Problems to solve at test, also theoretical: definitions, formulations of theorems. Similar practical and theoretical questions during exam.	K-W02++ K-W05++ K-U06++ K-U24+ K-K01+	P6S-KK P6S-UK P6S-UW P6S-WG P6S-WK

The syllabus of the module

Sem.	TK	The content	realized in	MEK
3	TK01	Notion of a metric space - examples. Balls, open sets, interior and boundary points of a set. Closed sets, the closure of a set, properties. A subspace of a metric space, open and closed sets in a subspace. A Cartesian product of metric spaces. Sequences of points, convergent sequences, properties.	W01 - W05, C01 -C05	MEK01 MEK04
3	TK02	mappings of metric spaces, continuous mappings, homeomorphism, uniformly continuous mappings. Compact metric spaces, equivalent conditions, properties of continuous functions on compact spaces.	W06 - W10, C06 - C10	MEK03
3	TK03	Cauchy sequences, complete spaces, Theorems of Cantor , Banach and Baire . Connected spaces, connected components of a space, arcwise connected spaces. Topological spaces, Hausdorff spaces.	W11 - W15, C11 - C15	MEK02 MEK04

The student's effort

The type of classes	The work before classes	The participation in classes	The work after classes
Lecture (sem. 3)	The preparation for a test: 8.00 hours/sem.	contact hours: 30.00 hours/sem.	complementing/reading through notes: 15.00 hours/sem. Studying the recommended bibliography: 5.00 hours/sem.
Class (sem. 3)	The preparation for a Class: 6.00 hours/sem. The preparation for a test: 6.00 hours/sem.	contact hours: 30.00 hours/sem.	Finishing/Studying tasks: 3.00 hours/sem. Others: 7.00 hours/sem.
Advice (sem. 3)	The preparation for Advice: 3.00 hours/sem.	The participation in Advice: 3.00 hours/sem.
Exam (sem. 3)	The preparation for an Exam: 10.00 hours/sem.	The written exam: 2.00 hours/sem.

The way of giving the component module grades and the final grade

The type of classes	The way of giving the final grade
Lecture	No grade for participation in the lecture.
Class	Passing grade from exercises based on grades from two tests. In the border case the activity during the execises may tip the scale.
The final grade

Sample problems

Required during the exam/when receiving the credit
(-)

Realized during classes/laboratories/projects
(-)

Others
(-)

Can a student use any teaching aids during the exam/when receiving the credit : no

The contents of the module are associated with the research profile yes

1	D. Wajnryb	The braid group and its presentation	2021