Topology
LEARNING MODULE DESCRIPTION (SYLLABUS)
TOPOLOGY
I. General information
1. Module title
Topology
2. Module code
DTOP UN0
3.Module type – compulsory or optional
optional
4.Programme title
Mathematics
5.Cycle of studies (1st or 2nd cycle of studies or full master’s programme)
2^{nd} cycle
6.Year of studies (where relevant)
First year of the second level
7.Terms in which taught (summer/winter term)
Winter term
8.Type of classes and the number of contact hours
Lecture 30 hours and practical classes 30=60 hours
9.Number of ECTS credits 6
10. Name, surname, academic degree/title of the module lecturer/other teaching staff/ email
Prof. dr hab. Jerzy Kąkol, kakol@amu.edu.pl
11. Language of classes
English
II. Detailed information
1. Module aim (aims)
General Topology with the introduction to the theory of homotopy is an essential part of mathematics, necessary to continue the studies in several areas of mathematics, for example analysis, functional analysis, theory of measure and probability. The aim is to provide the basic concepts, fundamental theorems and applications to analysis.
2. Prerequisites in terms of knowledge, skills and social competences (where relevant)
Basics of analysis and elementary set theory.
3. Module learning outcomes in terms of knowledge, skills and social competences and their reference to programme learning outcomes
Learning outcomes symbol* 
Upon completion of the course, the student will: 
Reference to programme learning outcomes^{#} 
…E_01 
be able to apply the basic concepts about elementary topology, like bases, subbases, fundamental properties of these concepts, theorems about (sequential or countable) compactness several concepts related with connected and locally connected spaces, locally compact spaces, continuity and sequential continuity, upper (lower) semicontinuity between abstractive topological spaces. 
K_MAT2_K01, KMAT2_U14, KMAT2_W02 
…E_02 
be able to apply Tychonoff’s theorem, fundamental metrization theorems to construct several mathematical objects, Stone’aCech compactifications and applications for the selected topics in (functional) analysis. This should provide technics to recognize classes of topological spaces which are homeomorphic. Students should be able to use Baire’s theorem to provide several examples of topological spaces with specific properties, they should know how to apply mentioned topological concepts to study the topology of spaces of continuous functions C(X) over compacta X and, more general, over Tychonoff spaces X, with the pointwise and compactopen topology, respectively. Theorems of Nagata, Ascoli, StoneWeierstrass about spaces C(X) will be presented and proved with possible applications. Students should recognize differences between metric complete spaces, completely metrizable spaces, Polish spaces, Cechcomplete spaces 
KMAT2_U08, KMAT2_U14 
…E_03 
be able to use and recognize the importance of several separation axioms, as well as, theorems which describe such concepts. Students will know the Urysohn’s theorem with possible applications, theorems dealing with problems about extensions of maps, like Tietze and Dugundji’s extensions theorems. Retracts and extenders will be discussed. 
KMAT2_U02, KMAT2_U18, KMAT2_W08 
E_04 
be able to use homotopic maps. 
KMAT2_U18

E_05 
be able to verify the difference between connected and arcwise connected spaces. 
KMAT2_U02 
E_06 
be able to explain the concept of the fundamental group (or first homotopy group) of the pair (X, x), where X is a topological space and x an element of X. 
KMAT_2W08, KMAT_2U18 
* module code, e.g. KHT_01 (KHT – module code in USOS; stands for Polish “Kataliza Heterogeniczna” /Heterogeneous Catalysis/ )
^{#} programme learning outcomes (e.g. K_W01, K_U01, … ); first K stands for programme title symbol in Polish, W for “wiedza” (knowledge) in Polish, U – for “umiejętności” (skills) in Polish, K – for “kompetencje społeczne” (social competences) in Polish
01, 02…  learning outcome number
4. Learning content
Module title 

Learning content symbol* 
Learning content description 
Reference to module learning outcomes^{ #} 
TK_01 
Concept of a topological space, open sets, closed sets, interior, closure, cluster points, base, subbase, several methods to construct topological spaces, for example topology given by metric spaces, examples and motivations from analysis. 
E_01 
TK_02 
Compact, countable compact, sequential compact spaces, differences between such concepts, examples. Connected spaces, locally connected spaces. Theorem characterizing those concepts with applications to analysis. 
E_01 
TK_3 
Topological products, Tychonoff’s theorem, applications. The StoneCech compactification, examples including the StoneCech compactification of the space of naturals numbers, applications to the theory of continuous functions C(X) over compacta X.

E_01, E_02 
TK_4 
Axioms of separation, Urysohn’s theorem with applications, Tietze and Dugundji’s extension theorems and applications. Spaces satisfying the first and the second axioms of countability. Urysohn’s metrization theorem, Some informations about Kakutani’s metrization theorems, sigmalocally finite families, networks and applications. 
E_02, E_03 
TK_5 
Locally compact spaces, Alexandrov’s compactification. Completely regular (=Tychonoff) spaces. Spaces Cc(X) and Cp(X) over Tychonoff spaces X endowed with the compactopen and pointwise topology, respectively. Nagata’s theorem about homeomorphic spaces Cp(X).

E_03, 
TK_6 
The concept of homotopy, homotopic maps, examples, fundamental theorems. 
E_04 
TK_7 
Arcwise connected spaces and their importance in the theory of homotopy, contractible spaces. 
E_05 
TK_8 
The first homotopy group for spaces (X, x) 
E_05, E_06 
* e.g. TK_01, TK_02, … (TK stands for “treści kształcenia” /learning content/ in Polish)
^{#} e.g. KHT_01 – module code as in Table in II.3
5. Reading list
N. Bourbaki, General Topology, Paris, Hermann 1966
R. Engelking, General Topology PWN Warsaw (Polish Edition) 1977
J. Kelley, General Topology, Van Nostrand Company 1955.
J. Munkres, Topology, PrenticeHall 1974.
W. Rudin, Functional Analysis PWN Warsaw (Polish Edition) 2001
I. M. Singer, J. A Thorpe Lecture Notes on Elementary Topology and Geometry, Undegraduate Texts in Mathematics, Springer 1967.
6. Information on the use of blendedlearning (if relevant)
7. Information on where to find course materials
In the library
III. Additional information
1. Reference of learning outcomes and learning content to teaching and learning methods and assessment methods
Module title 

Symbol of module learning outcome* 
Symbol of module learning content^{#} 
Methods of teaching and learning 
Assessment methods of LO achievement^{&} 
E_01 
T_01, T_02, T_03 
Lectures, solving exercises in a group 
Final Exam, Test 
E_02 
T_03, T_04 
Lectures and exercises in a a group 
Final Exam, Test 
E_03 
T_03, T_05 
Lecture, solving exercises in a group 
Final Exam 
E_04 
T_06 
Lecture, solving exercises in a group 
Final Exam 
E_05 
T_07, T08 
Lecture, exercises in a group 
Final Exam 
E_06 
T_08 
Lecture, exercises in a group 
Final Exam 
* e.g. KHT_01 – module code as in Table in II.3 and II.4
^{#} e.g. TK_01 – learning content symbol as in II.4
^{&} Please include both formative (F) and summative (S) assessment
It is advisable to include assessment tasks (questions).
2. Student workload (ECTS credits)
Module title: 

Activity types 
Mean number of hours* spent on each activity type 
Contact hours with the teacher as specified in the programme 
60 
Independent study 1 
30 




Total hours 
90 
Total ECTS credits for the module 
6 
* Class hours – 1 hour means 45 minutes
^{#}Independent study – examples of activity types: (1) preparation for classes, (2) data analysis, (3) librarybased work, (4)writing a class report, (5) exam preparation, etc.
3. Assessment criteria
4. Titles of classes
Syllabus: 

Week 1 
The concept of topology and fundamental facts, examples. 
Week 2 
Compact and connected spaces. 
Week 3 
Tychonoff’s theorem and applications. 
Week 4 
The StoneCech compactification I. 
Week 5 
The StoneCech compactification II, applications. 
Week 6 
Axioms of separation. 
Week 7 
Urysohn’s theorem, applications. 
Week 8 
Tietze, Dugundji’s extension theorems. 
Week 9 
Locally compact spaces, Alexandrov’s theorem, Tychonoff spaces. 
Week 10 
Topology of spaces Cc(X) and Cp(X). 
Week 11 
Ascoli’s theorem, StoneWeierstrass theorem 
Week 12 
Nagata’s theorem about spaces Cp(X). 
Week 13 
Homotopy, homotopic maps. 
Week 14 
Contractible spaces. 
Week 15 
The fundamental group. 