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Applications of descriptive set theory in mathematical analysis
Doležal, Martin ; Zelený, Miroslav (advisor) ; Holický, Petr (referee) ; Zapletal, Jindřich (referee)
We characterize various types of σ-porosity via an infinite game in terms of winning strategies. We use a modification of the game to prove and reprove some new and older in- scribing theorems for σ-ideals of σ-porous type in locally compact metric spaces. We show that there exists a closed set which is σ-(1 − ε)-symmetrically porous for every 0 < ε < 1 but which is not σ-1-symmetrically porous. Next, we prove that the realizable by an action unitary representations of a finite abelian group Γ on an infinite-dimensional complex Hilbert space H form a comeager set in Rep(Γ, H). 1
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Properties of sigma-porous sets
Rmoutil, Martin ; Zajíček, Luděk (advisor) ; Zelený, Miroslav (referee)
In the present thesis we prove several new results concerning -porous sets. In the first two chapters we examine some properties of related sets in the space R while in the third chapter we concentrate on an entirely different problem formulated in the setting of topologically complete metric spaces. To be more specific, in the first chapter we prove non- -porosity of the set Ad of all real numbers x (0, 1) with decimal expansion containing the number 9 with density d. In spite of being relatively difficult, this new result has little importance in itself. It merely answers a natural question which arises from an article of L. Zajíček [8]. The main result presented in the second chapter is a significant improve- ment of the following result of R.J. Najáres and L. Zajíček from the article [5]: There exists a closed set F R which is right porous, but is not -left porous. Thus for any kind of "upper" porosity (i.e. a porosity defined using limsup) it is now even more unlikely for any connection between "left" and "right" to be discovered. From another work [10] of L. Zajíček arises the following question: If A X and B Y are two non- -lower porous G -subsets of topologically complete metric spaces X and Y , is it necessarily true that the Cartesian product A × B is also non- -lower porous? The article [10]...
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Collections of compact sets in descriptive set theory
Vlasák, Václav ; Zelený, Miroslav (advisor) ; Holický, Petr (referee) ; Tišer, Jaroslav (referee)
1 Title: Collections of compact sets in descriptive set theory Author: Václav Vlasák Department: Department of Mathematical Analysis Supervisor: Doc. RNDr. Miroslav Zelený, Ph.D. Author's e-mail address: vlasakmm@volny.cz Abstract: This work consists of three articles. In Chapter 2, we dissert on the connections between complexity of a function f from a Polish space X to a Polish space Y and complexity of the set C(f) = {K ∈ K(X); f K is continuous}, where K(X) denotes the space of all compact subsets of X equipped with the Vietoris topology. We prove that if C(f) is analytic, then f is Borel; and assuming ∆1 2-Determinacy we show that f is Borel if and only if C(f) is coanalytic. Similar results for projective classes are also presented. In Chapter 3, we continue in our investigation of collection C(f) and also study its restriction on convergent sequences (C(f)). We prove that C(f) is Borel if and only if f is Borel. Similar results for projective classes are also presented. The Chapter 4 disserts on HN -sets, which form an important subclass of the class of sets of uniqueness for trigonometric series. We investigate the size of these classes which is reflected by the family of measures called polar which annihilate all the sets belonging to the given class. The main aim of this chapter is to answer in...
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