1. SylabUZ
2. Faculty of Mathematics, Computer Science and Econometrics
3. winter term 2019/2020
4. Mathematics - First-cycle studies leading to Bachelor's degree
5. Topology

Generate PDF for this page

Topology - course description

Aim of the course

Student should be familiar with the basic concepts of topology metric spaces: metric space, convergence in metric spaces, limit and continuity of mappings between metric spaces, separable, compact, complete and connected metric spaces.

Prerequisites

Save changes

Standard graduate courses in the set theory and mathematical analysis.

Scope

Lecture
Metric spaces
1. Elementary properties and examples of metric spaces. Function spaces.(2 hours)
2. The topology defined by a metric. Base of a metric space. System of neighborhoods. Interior and closure of sets. Open and closed sets.(2 hours)
3. Convergence of sequences in metric spaces. Comparison of metrics.(1hour)
4. Subspace of a metric space. Cartesian product of metric spaces.(2 hours)
5. Different sets in metric spaces.(1 hour)
6. Separable metric spaces - basic properties and examples.(1 hour)
Continuous mappings between metric spaces
1. Continuous mappings and their characterizations. Uniformly continuous mappings.(2 hours)
2. Homeomorphisms and isometries between metric spaces. Topological invariants.(1 hour)
3. Convergence of sequences of functions.
Complete metric spaces
1. Complete metric spaces. Elementary properties and examples.(2 hours)
2. Completion of metric spaces.(1 hour)
3. Baire category theorem. Baire category method.(1 hour)
4. Banach fixed point theorem.(1 hour)
Compact metric spaces
1. Compact metric spaces. Elementary properties and examples.(2 hours)
2. Characterizations of compact metric spaces. Borel-Lebesgue theorem.(2 hours)
3. Cartesian product of compact spaces.(1 hour)
4. Characterization of compact sets in Euclidean spaces.(1 hour)
5. Properties of continuous mappings on compact metric spaces. Weierstrass theorem.(3 hours)
Connected and arc connected metric spaces
1. Connected metric spaces. Elementary properties and examples.(1 hour)
2. Properties of continuous mappings on connected metric spaces.(1 hour)

Class
Metric spaces
1. Elementary properties of metrics. Euclidean spaces and function spaces.(2 hours)
2. Examining of metric conditions in concrete function spaces.(3 hours)
3. Comparison of metrics on the plane.(2 hours)
4. Examining of Cartesian products of metric spaces.(2 hours)
5. Operations on sets in metric spaces : calculation of the interior and the closure of sets in metric spaces.(4 hours)
6. Examining of convergence and of sequences in metric spaces.(2 hours)
7. Determination of different classes of sets in metric spaces.(2 hours)
8. Colloquium.(2 hours)
Continuous mappings
1. Examining of continuity and uniform continuity of functions on function spaces.(4 hours)
2. Examining of convergence of sequences in function spaces.(2 hours)
Topological properties basic classes of metric spaces
1. Examining of completeness of function metric spaces.(2 hours)
2. Characterization of compact and connected sets in metric spaces.(3 hours)
3. Colloquium.(2 hours)

Teaching methods

Traditional lecture, open to discussion; classes with lists of exercises and problems to be solved by students.

Learning outcomes and methods of theirs verification

Outcome description	Outcome symbols	Methods of verification	The class form

Assignment conditions

1. Verification of preparation of students and their activity during classes.
2. Colloquia with various degree of difficulties, allowing to verify if students attained learning outcomes.
3. Exam (written) checks the understanding of the basic notions, examples and proofs of theorems basing on the indicated earlier examination criteria.
Passing the exam: the weighted mean of notes of the classes (40%) and the exam (60%).
A positive note of the classes is the necessary condition to be admitted to the exam.
A positive note of the exam attests the subject.

Recommended reading

1. K. Janich, Topology, Springer-Verlag, New York Berlin Heidelberg Tokyo, 1984.
2. S. Gładysz, Wstęp do topologii, Wydawnictwo Naukowe PWN, Warszawa 1981.
3. W. Rzymowski, Przestrzenie metryczne w analizie, Wyd. UMSC, Lublin 2000.
4. J. Jędzejewski, W. Wilczyński, Przestrzenie metryczne w zadaniach , Wyd. UŁ. Łódz 2007.

Further reading

1. J. Jędrzejewski, Zarys teorii przestrzeni metrycznych, Wydawnictwo WSP Słupsk, 1999.
2. W. Archangielski, W.I. Ponomariow, Podstawy topologii ogólnej w zadaniach, PWN, Warszawa 1986.

Notes

Modified by dr Robert Dylewski, prof. UZ (last modification: 20-09-2019 10:18)