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Course, academic year 2018/2019
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Theory of Real Functions 2 - NMMA404
Title in English: Reálné funkce 2
Guaranteed by: Department of Mathematical Analysis (32-KMA)
Faculty: Faculty of Mathematics and Physics
Actual: from 2018 to 2018
Semester: summer
E-Credits: 4
Hours per week, examination: summer s.:2/0 Ex [hours/week]
Capacity: unlimited
Min. number of students: unlimited
State of the course: taught
Language: Czech, English
Teaching methods: full-time
Guarantor: prof. RNDr. Luděk Zajíček, DrSc.
Class: M Mgr. MA
M Mgr. MA > Povinné
Classification: Mathematics > Real and Complex Analysis
Incompatibility : NRFA013
Interchangeability : NRFA013
Annotation -
Last update: T_KMA (02.05.2013)
Mandatory course for the master study branch Mathematical analysis. Recommended for the first year of master studies. Content: topological properties of complete metric spaces, Borel sets and functions, analytic sets.
Course completion requirements -
Last update: prof. RNDr. Luděk Zajíček, DrSc. (08.05.2018)

The exam will be oral, approximately 60 minutes long. The exam examines the knowledge of the subject matter in the scope given by the lectures.

Literature -
Last update: T_KMA (25.04.2013)

Kechris, Alexander S.: Classical descriptive set theory. Graduate Texts in Mathematics, 156. Springer-Verlag, New York, 1995.

Oxtoby, John C. Measure and category. A survey of the analogies between topological and measure spaces. Second edition. Graduate Texts in Mathematics, 2. Springer-Verlag, New York-Berlin, 1980

Requirements to the exam - Czech
Last update: doc. RNDr. Miroslav Zelený, Ph.D. (11.10.2017)

Zkouška bude ústní a bude testována znalost definic a vět,

které byly v kurzu odpředneseny.

Syllabus -
Last update: T_KMA (16.09.2013)

1. Topological properties of separable complete metric spaces

  • Cantor-Bendixson theorem
  • embedding of the Cantor set into a complete space
  • sets with the Baire property
  • Bernstein nonmeasurable sets
  • irrational numbers as $N^N$

2. Borel functions and sets

  • semicontinuous functions
  • functions of the first Baire class
  • elements of the Borel hiearchy

3. Analytic sets

  • stability to set operations, Suslin scheme
  • relationship to Borel sets
  • measurability and Baire property
  • embedding of the Cantor set

Entry requirements -
Last update: prof. RNDr. Luděk Zajíček, DrSc. (08.05.2018)

Theory of metric spaces, as covered by lectures NMMA201 and NMMA202.

 
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