Point-set topology

2022/2023
Programme:
Mathematics, First Cycle
Year:
2 year
Semester:
first
Kind:
mandatory
ECTS:
5
Language:
slovenian
Hours per week – 1. semester:
Lectures
2
Seminar
0
Tutorial
2
Lab
0
Prerequisites

Completed course Analysis 1.

Content (Syllabus outline)

Topological spaces, confinuous mappings, homeomorphisms. Bases and subbases. Product topology, mappings into products. Subspaces, embeddings, piecewise definition of mappings. Hereditary and multiplicative properties.

Separation axioms, connectedness and path-connectedness, components, local-connectedness. Compactness, local compactness, compactification. Baire theorem, Cantor set.

Mapping spaces, compact-open topology. Mappings on normal spaces, Urysohn lemma, Tietze theorem, Stone-Weierstrass theorem. Urysohn metrization theorem, retracts and extensors, partitions of unity.

Readings

Osnovna:
J. Mrčun: Topologija, (Izbrana poglavja iz matematike in računalništva, 44). Ljubljana: DMFA - založništvo, 2008. VI, 147 str., ilustr. ISBN 978-961-212-207-2.

P. Pavešić: Splošna topologija, (Izbrana poglavja iz matematike in računalništva, 43). 2. natis. Ljubljana: DMFA - založništvo, 2017. VI, 89 str., ilustr. ISBN 978-961-212-205-8..

J. Smrekar: Rešene naloge iz splošne topologije, (Izbrana poglavja iz matematike in računalništva, 50). Ljubljana: DMFA - založništvo, 2017. VI, 119 str., ilustr. ISBN 978-961-212-285-0.

Zapiski predavanj in vaj

Dodatna:
J. Dugundji: Topology. Allyn and Bacon, Inc., Boston, Mass. 1966 xvi+447 pp

J. R. Munkres: Topology. Prentice Hall, Inc., Upper Saddle River, NJ, 2000. xvi+537 pp. ISBN: 0-13-181629-2

N. Prijatelj: Matematične strukture. 3, Okolice, (Knjižnica Sigma, 23.a.). 2. natis. Ljubljana: Državna založba Slovenije, 1985. 264 str. ISBN 961-212-176-1.

M. Cencelj, D. Repovš: Topologija, (Zbirka Pitagora). 1. ponatis. Ljubljana: Pedagoška fakulteta, 2011. XVI, 169 str., ilustr. ISBN 978-86-7735-051-2.

PAVEŠIĆ, Petar, VAVPETIČ, Aleš. Rešene naloge iz topologije, (Izbrana poglavja iz matematike in računalništva, 32). 2. izd. Ljubljana: Društvo matematikov, fizikov in astronomov Slovenije, 1997. 141 str. ISBN 961-212-079-X

Objectives and competences

Student gets familiar with basic concepts point-set topology, such as connectedness, compactness, separation properties, topology on products and function spaces.

Intended learning outcomes

Knowledge and understanding: Understanding of notions such as topology, continuous map, connectedness and compactnes. Knowledge of basic concepts of the above notions and connection with other areas of mathematics.
Application: Point-set topology is one of the basic mathematical courses. Student gets familiar with basic definitions and techniques that are fondations for several other mathematical courses.
Reflection: Understanding of the theory from the applications.
Transferable skills: The ability to formualate a problem in suitable language, find a solution of the problems and analyse the method on real examples.

Learning and teaching methods

Lectures, exercises, homework, consultations

Assessment

Practical and theoretical knowledge are graded separately.

Practical knowledge is assessed based on two tests (15% of the grade each) and the final exercise based exams (70% of the grade).

Theoretical knowledge is normally assessed as an oral exam.

grading: 5 (fail), 6-10 (pass) (according to the Statute of UL)

Lecturer's references

Petar Pavešić:
– PAVEŠIĆ, Petar CONNER, Gregory R., HERFORT, Wolfgang, PAVEŠIĆ, Petar. Some anomalous examples of lifting spaces. Topology and its Applications, ISSN 0166-8641. [Print ed.], April 2018, vol. 239, str. 234-243.
– PAVEŠIĆ, Petar, A topologist's view of kinematic maps and manipulation complexity. V: GRANT, Mark (ur.). Topological complexity and related topics : Mini-Workshop Topological Complexity and Related Topics, February 28 - March 5, 2016, Mathematisches Forschungsinstitut Oberwolfach, Oberwolfach, Germany, (Contemporary mathematics, ISSN 0271-4132, 702). Providence: American Mathematical Society.
– PAVEŠIĆ, Petar. Splošna topologija, (Izbrana poglavja iz matematike in računalništva, 43). Ljubljana: DMFA - založništvo, 2008. VI, 89 str., ilustr. ISBN 978-961-212-205-8 [COBISS-SI-ID 240425984]

Dušan Repovš:
– BANAKH, Taras, REPOVŠ, Dušan. Direct limit topologies in the categories of topological groups and of uniform spaces. Tohoku mathematical journal, ISSN 0040-8735, 2012, vol. 64, no. 1, str. 1-24 [COBISS-SI-ID 16215897]
– CÁRDENAS, Manuel, LASHERAS, Francisco F., QUINTERO, Antonio, REPOVŠ, Dušan. On manifolds with nonhomogeneous factors. Central European Journal of Mathematics, ISSN 1895-1074, 2012, vol. 10, no. 3, str. 857-862 [COBISS-SI-ID 16241753]
– KARIMOV, Umed H., REPOVŠ, Dušan. On generalized 3-manifolds which are not homologically locally connected. Topology and its Applications, ISSN 0166-8641. [Print ed.], 2013, vol. 160, iss. 3, str. 445-449 [COBISS-SI-ID 16558681]
– CENCELJ, Matija, REPOVŠ, Dušan. Topologija, (Zbirka Pitagora). 1. ponatis. Ljubljana: Pedagoška fakulteta, 2011. XVI, 169 str., ilustr. ISBN 978-86-7735-051-2 [COBISS-SI-ID 254230528]