Topology I

General data

Course ID:	1000-113bTP1b
Erasmus code / ISCED:	11.1 Kod klasyfikacyjny przedmiotu składa się z trzech do pięciu cyfr, przy czym trzy pierwsze oznaczają klasyfikację dziedziny wg. Listy kodów dziedzin obowiązującej w programie Socrates/Erasmus, czwarta (dotąd na ogół 0) – ewentualne uszczegółowienie informacji o dyscyplinie, piąta – stopień zaawansowania przedmiotu ustalony na podstawie roku studiów, dla którego przedmiot jest przeznaczony. / (0541) Mathematics The ISCED (International Standard Classification of Education) code has been designed by UNESCO.
Course title:	Topology I
Name in Polish:	Topologia I (potok 2)
Organizational unit:	Faculty of Mathematics, Informatics, and Mechanics
Course groups:	Obligatory courses for 2nd grade JSEM Obligatory courses for 2nd grade JSIM (3I+4M) Obligatory courses for 2nd grade JSIM (3M+4I) Obligatory courses for 2rd grade Mathematics
ECTS credit allocation (and other scores):	(not available) Basic information on ECTS credits allocation principles: the annual hourly workload of the student’s work required to achieve the expected learning outcomes for a given stage is 1500-1800h, corresponding to 60 ECTS; the student’s weekly hourly workload is 45 h; 1 ECTS point corresponds to 25-30 hours of student work needed to achieve the assumed learning outcomes; weekly student workload necessary to achieve the assumed learning outcomes allows to obtain 1.5 ECTS; work required to pass the course, which has been assigned 3 ECTS, constitutes 10% of the semester student load. view allocation of credits
Language:	Polish
Type of course:	obligatory courses
Short description:	This course presents basic notions of topology: metric and topological spaces, continuous maps, homeomorphisms, Cartesian products, complete metric spaces, compactness, connectedness and path connectedness, homotopy of maps and loops, contractibility, quotient spaces.
Full description:	1. Metric spaces. Topology of metric spaces. Topological spaces. Base of a topology. Interior and closure of a set, subspaces. Hausdorff spaces. Continuous mappings, characterizations of continuity. Homeomorphisms. Tietze Theorem on extensions of mappings (for metrizable spaces). Cartesian products of topological spaces. Separable spaces. (3 lectures) 2. Compact spaces. Conditions equivalent to compactness in metrizable spaces. Compact subsets of Euclidean spaces. Continuous mappings on compact spaces. Weierstrass Theorem. A continuous bijection from a compact space to a Hausdorff space is a homeomorphism. Uniform continuity. The Cantor set. Tychonoff Theorem on compactness of Cartesian products of compact spaces (proof for finite products). (3 lectures) 3. Complete metric spaces. If a metric space $Y$ is complete then the space of bounded continuous functions $C_{b}(X,Y)$ equipped with the sup metric is complete. Banach Fixed Point Theorem. Baire Theorem. Metric space is compact iff it is complete and totally bounded. Ascoli-Arzeli Theorem. (2 lectures) 4. Connected spaces. Path connectedness. Components and path components. (1 lecture) 5. Homotopic mappings. Contractible spaces. Homotopic loops. Simply connected spaces. Proof of the noncontractibility of the circle. Corollaries: there is no retraction from a disc onto its boundary circle, Brouwer Fixed Point Theorem for dimension 2. Proof of the Fundamental Theorem of Algebra. (3 lectures) 6. Quotient spaces. Attaching a space $Y$ to $X$ along $A \subset Y$ via $f : A \to X$. Two-dimensional manifolds. Examples of surfaces obtained by identification of edges of regular polygons. (2 lectures).
Bibliography:	1. J. Dugundji, Topology, Allyn and Bacon, Inc., Boston, 1966. 2. R. Engelking, K. Sieklucki, Topology. A Geometric Approach, Heldermann Verlag, Berlin, 1992. 3. K. Janish, Topology, Springer Verlag, New York, 1990.

This course is not currently offered.

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