Point-set topology

2018/2019
Programme:
Mathematics Education
Year:
2 year
Semester:
first
Kind:
mandatory
ECTS:
5
Language:
slovenian
Lecturer (contact person):
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

J. Dugundji: Topology.

J. R. Munkres: Topology : A First Course.

J. Mrčun: Topologija, zapiski predavanj.

P. Pavešić: Splošna topologija.

N. Prijatelj: Matematične strukture III : Okolice.

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

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

Lecturer's references

Janez Mrčun:
KALIŠNIK, Jure, MRČUN, Janez. A Cartier-Gabriel-Kostant structure theorem for Hopf algebroids. Advances in mathematics, ISSN 0001-8708, 2013, vol. 232, iss. 1, str. 295-310. [COBISS-SI-ID 16432473]
KALIŠNIK, Jure, MRČUN, Janez. Equivalence between the Morita categories of étale Lie groupoids and locally grouplike Hopf algebroids. Indagationes mathematicae, ISSN 0019-3577, 2008, vol. 19, no. 1, str. 73-96. [COBISS-SI-ID 14978393]
MRČUN, Janez. Topologija, (Izbrana poglavja iz matematike in računalništva, 44). Ljubljana: DMFA - založništvo, 2008. VI, 147 str., ilustr. ISBN 978-961-212-207-2. [COBISS-SI-ID 243021824]
MOERDIJK, Ieke, MRČUN, Janez. Introduction to foliations and Lie groupoids, (Cambridge studies in advanced mathematics, 91). Cambridge, UK: Cambridge University Press, 2003. IX, 173 str., ilustr. ISBN 0-521-83197-0. [COBISS-SI-ID 12683097]
Petar Pavešić:
PAVEŠIĆ, Petar. A note on trivial fibrations. Glasnik matematički. Serija 3, ISSN 0017-095X, 2011, vol. 46, no. 2, str. 513-519. [COBISS-SI-ID 16078681]
PAVEŠIĆ, Petar. Decompositions of groups of invertible elements in a ring. Proceedings. Section A, Mathematics, ISSN 0308-2105, 2009, vol. 139, iss 6, str. 1275-1287. [COBISS-SI-ID 15505497]
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]
PAVEŠIĆ, Petar. Rešene naloge iz topologije, (Izbrana poglavja iz matematike in računalništva, 32). Ljubljana: Društvo matematikov, fizikov in astronomov Slovenije, 1995. 132 str. ISBN 961-212-042-0. [COBISS-SI-ID 47811328]
Dušan Repovš:
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]
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]
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]
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]