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
01	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
02	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
03	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
04	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

Attention: Depending on the epidemic situation, verification of the achieved learning outcomes specified in the study program, in particular credits and examinations at the end of specific classes, can be implemented remotely (real-time meetings).

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	Final grade based on the final exam. Passing grade form the exercises is the condition to take the exam. In the border case good grade from the exercises or a few oral questions may tip the scale by one half of a unit (say from 4 to 4,5).

Sample problems

Required during the exam/when receiving the credit
ExamTop2011.pdf
ExamTopKomis2010.pdf

Realized during classes/laboratories/projects
koloTopZal2009.pdf
koloIITop2011.pdf

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