|
| |
|
|
Generalized limits of affine functions
Holub, Aleš ; Spurný, Jiří (advisor) ; Holický, Petr (referee)
We construct a co-analytic filter on the set of finite sequences of natural numbers, which allows us to obtain a strongly affine function of arbitrary Borel class from compact convex subset of locally convex space through single limit process (by this filter) applied to countable system of affine continuous functions. Conversely we show that function obtainted as result of such process is necessarily Borel and strongly affine. Further we generalize this method using metrizable reduction approach for Baire functions on non-metrizable spaces. Last chapter covers similar result for bi-analytic functions on separable metrizable spaces.
|
|
|
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...
|
|
|
Properties of Cantor function
Fiala, Martin ; Hencl, Stanislav (advisor) ; Holický, Petr (referee)
Properties of Cantor function Author: Martin Fiala Supervisor: Stanislav Hencl Abstract: In the present thesis we study main properties of the Can- tor function (sometimes called Cantor Devil's staircase in popular lit- erature), named after significant german mathematician Georg Cantor ( 3 March 1845 in St Petersburg, 6 Jan 1918 in Halle). 1
|
|
| |
|
|
Borel sets in topological spaces
Vondrouš, David ; Spurný, Jiří (advisor) ; Holický, Petr (referee)
This thesis deals with study of mappings preserving Borel classes or absolute Borel classes. We prove a theorem which shows that under some assumptions there exists a (selection) function with certain properties. Using this theorem we obtain several results on preservation of Borel classes. Moreover, thanks to that theorem we prove a theorem on preservation of absolute Borel classes under a perfect mapping. Next, we show an assertion which implies that a piecewise closed mapping has a restriction that is "piecewise perfect" and its image is equal to the image of the original mapping. Under certain additional assumptions we prove a similar assertion for an Fσ-mapping instead of a piecewise closed mapping. Using these assertions and the theorem on preservation of absolute Borel classes under a perfect mapping we obtain further results on preservation of absolute Borel classes, in particular, for piecewise closed mappings and Fσ- -mappings. In the last chapter we study mappings such that the inverse image of an open set under these mappings is of a particular additive class. 1
|
|
|
Series in Banach spaces
Minasjan, Martin ; Spurný, Jiří (advisor) ; Holický, Petr (referee)
In this thesis we introduce several different types of series convergence in nor- med vector spaces and study relations between them. Furthermore, we will pro- ove the equivalence of all defined types of convergence in Banach spaces, we call this convergence unconditional convergence. Finally, we will show the Dvoretzky- Rogers theorem, i.e. that in all infinitely dimensional Banach spaces there is a series that is unconditionally convergent, but not absolutely convergent. 1
|
|
|
Absolute and non-absolute F-Borel spaces
Kovařík, Vojtěch ; Kalenda, Ondřej (advisor) ; Matheron, Ethienne (referee) ; Holický, Petr (referee)
We investigate F-Borel topological spaces. We focus on finding out how a complexity of a space depends on where the space is embedded. Of a particular interest is the problem of determining whether a complexity of given space X is absolute (that is, the same in every compactification of X). We show that the complexity of metrizable spaces is absolute and provide a sufficient condition for a topological space to be absolutely Fσδ. We then investigate the relation between local and global complexity. To improve our understanding of F-Borel spaces, we introduce different ways of representing an F-Borel set. We use these tools to construct a hierarchy of F-Borel spaces with non-absolute complexity, and to prove several other results. 1
|
|
|
Continuous mappings and fixed-point theorems
Vondrouš, David ; Holický, Petr (advisor) ; Zelený, Miroslav (referee)
This thesis deals with images of compact convex sets under a continuous mapping. We will show a combinatorial proof of famous Brouwer's fixed-point theorem based on Sperner's lemma. Later, this theorem will be applied for proving Brouwer's invariance of domain theorem, which asserts that image of an open subset of an euclidean space under a continuous mapping is open too. Then we will compare this proof with another proof using Borsuk's theorems. Their proof is more complicated, nevertheless it turns out that Borsuk's theorems give stronger results. One of them is, for instance, an analogy of the Darboux property for continuous mappings in an multidimensional space. 1
|
|
|
Sigma-ideal of sigma-porous sets
Hronek, Radek ; Zelený, Miroslav (advisor) ; Holický, Petr (referee)
This bachelor thesis deals with concepts of porous and σ-porous sets, where we prove some basic properties. First, we define terms on the real axis, in other chapters we generalize them into metric spaces. At the end of the thesis there are several interesting examples. In the first chapter we focus on demonstration that σ-porous sets are sets of the first category. The main result of the chapter is that the Lebesgue measure of the σ-porous sets in the space Rn is 0. In the following chapter we deal with the construction of certain sets in the space of continuous functions, in the second case in the space of the nonempty compact sets in Rn . In both cases we show that given sets are σ-porous.
|
guest ::
