1. SylabUZ
2. Wydział Matematyki, Informatyki i Ekonometrii
3. semestr zimowy 2022/2023
4. Mathematics - pierwszego stopnia z tyt. licencjata
5. Topology

Wygeneruj PDF dla tej strony

Topology - opis przedmiotu

Cel przedmiotu

Getting acquainted students with the basic concepts of metric space topology: metric space, convergence of sequence, limit and continuity of function, separability, compactness, completeness and connectedness of metric space.

Wymagania wstępne

Zapisz zmiany

Knowledge of basics of set theory and mathematical analysis.

Zakres tematyczny

Lecture

Metric spaces

     1. Basic properties and examples of a metric spaces. Functional spaces. (2 hours)          

     2. Topology generated by metric. Base of a metric space. Local base at a point. Interior    

         and closure of a set. Open and closed sets. (2 hours)

     3. Convergence of sequences in metric spaces. Comparing metrics. (1 hour)

     4. Subspaces of metric spaces. Cartesian product of metric spaces. (2 hours)

     5. Various types of sets in metric spaces. (1 hour)

     6. Separated metric spaces – basic properties and examples. (1 hour)

Continuous transformations of metric spaces

     1. Continuous transformations and their characterizations. Uniformly continuous transformations. (3 hours)

     2. Homeomorphisms and isometries of metric spaces. Topological invariants. (1 hour)                                 

     3. Convergence of function sequences. (1 hour)

Complete metric spaces

     1. Complete spaces. Basic properties and examples. (2 hours) 

     2. Completion of metric spaces. (1 hour)

     3. Baire category theorem. Method of Baire category. (1 hour)   

     4. Banach fixed point theorem. (1 hour).

Compact metric spaces

     1. Compact spaces. Basic 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 space. (1 hour)

     5. Properties of continuous transformations on compact metric spaces. Weierstrass theorem. (3 hours). 

Connected and arcwise connected metric spaces

     1. Connected spaces. Basic properties and examples. (1 hour)

     2. Properties of continuous transformations on connected metric spaces. (1 hour).

 

Exercises

Metric spaces

     1. Basic properties of metrics. Euclidean spaces and function spaces. (2 hours)

     2. Checking metric conditions in specific function spaces. (3 hours)

     3. Comparing various metrics on plane. (2 hours)

     4. Study of Cartesian products of metric spaces. (2 hours)

     5. Operations on sets in a metric space, e.g. determining the interior and closer of sets for various metrics.

         (4 hours)

     6. Study of convergence and accumulation points of sequences in metric spaces. (2 hours)

     7. Determination of various properties of sets in metric spaces. (2 hours)

     8. Colloquium (2 hours).

Continuous transformations

     1. Study of continuity and uniform continuity of functions in function spaces. (4 hours)

     2. Study of convergence in function spaces (2 hours).

Topological properties of basic classes of metric spaces

     1. Study of completeness of function spaces. (2 hours)

     2. Characterization of compact and connected sets in metric spaces. (3 hours)

     3. Colloquium (2 hours).

 

Metody kształcenia

Conventional lecture. Auditorium exercises, solving problems and tasks.

Efekty uczenia się i metody weryfikacji osiągania efektów uczenia się

Opis efektu	Symbole efektów	Metody weryfikacji	Forma zajęć

Warunki zaliczenia

     1. Colloquia of varying difficulty, allowing to check the degree of mastery of individual learning outcomes.
     2. The exam consists in checking the understanding of basic concepts, giving examples and checking the knowledge of proofs of the indicated claims.

The assessment of the subject consists of an assessment of the exercises (40%) and an assessment of the exam (60%).

The condition for taking the exam is a positive assessment of the exercises. The condition for passing the subject is a positive assessment of the exam.

Literatura podstawowa

1. K. Kuratowski, Topology, Vol.I, PWN - Polish Scientific Publishers, Warszawa, and   

    Academic Press, New York, 1966.

2. K. Kuratowski, Topology, Vol.II, PWN - Polish Scientific Publishers, Warszawa, and   

    Academic Press, New York, 1968.

Literatura uzupełniająca

1. John L. Kelley, General Topology, Grad. Texts in Math., No. 27, Springer-Verlag, New   

    York-Berlin, 1975, xiv+298 pp.

Uwagi

Zmodyfikowane przez prof. dr hab. Witold Jarczyk (ostatnia modyfikacja: 21-01-2024 23:30)