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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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Descriptive properties of sets in Banach spaces
Kurka, Ondřej ; Zajíček, Luděk (referee) ; Holický, Petr (advisor)
An essential part of the work is devoted to the study of the sets of Fréchet subdi erentiability from the view of the descriptive set theory. Proofs of the known results of L. Zajíek, P. Holický, M. Laczkovich and M. Šmídek are given. A new result is that there exists a Lipschitz function with non-Borel set of Fréchet subdi erentiability on every non-reflexive Banach space. The Borel classes of the sets of Fréchet subdi erentiability of continuous functions on reflexive spaces are studied as well. Further, some sets of sequences in Banach spaces are investigated. A modi ed proof of the theorem of R. Kaufman which says that every non-re exive Banach space can be renormed not to have Borel set of norm-attaining functionals is shown. A characterization of non-quasire exive Banach spaces is given.
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