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Complexity of classification problems in topology
Dudák, Jan ; Vejnar, Benjamin (advisor) ; Krupski, Pawel (referee) ; Zelený, Miroslav (referee)
This thesis consists of three articles. The first article focuses on compact metrizable spaces homeomorphic to their respective squares, the main result being that there ex- ists a family of size continuum of pairwise non-homeomorphic compact metrizable zero- dimensional spaces homeomorphic to their respective squares. This result answers a question of W. J. Charatonik. In the second article we prove that there exists a Borel measurable mapping assigning to each Peano continuum X a continuous function from [0, 1] onto X. We also show that there exists a Borel measurable mapping assigning to each triple (X, x, y), where X is a Peano continuum and x, y are distinct points in X, an arc in X with endpoints x, y. In the third article we prove that the homeomorphism relation for absolute retracts in R2 is Borel bireducible with the isomorphism relation for countable graphs. Moreover, we prove that neither the homeomorphism relation for Peano continua in R2 nor the homeomorphism relation for absolute retracts in R3 is clas- sifiable by countable structures. We also show that the homeomorphism relation (as well as the ambient homeomorphism relation) for compacta in [0, 1]n is Borel reducible to the homeomorphism relation for continua in [0, 1]n+1 . 1
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Images of Typical Continuous Functions
Nešvera, Michal ; Vejnar, Benjamin (advisor) ; Holický, Petr (referee)
It follows from the Baire theorem that comeagre sets in complete metric spaces are "topologically large". Properties that are satisfied by a large set are called typical. The proofs of statements concerning typical properties of continuous functions are the main part of this work. For this purpose, the necessary definitions are introduced in the first chapter and the completeness of spaces of continuous functions is proved. As the first example of a typical property, in the second chapter we prove the Banach-Mazurkiewicz theorem, which states that non-differentiability is a typical property. The third chapter of this thesis is devoted to the study of typical properties of continuous mappings of the unit interval into the plane. In the last chapter, statements regarding the typical properties of continuous mappings of the unit interval into Euclidean spaces of higher dimensions are proved. 1
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Brouwer fixed point theorem (proofs and history)
Vítek, Tomáš ; Hušek, Miroslav (advisor) ; Vejnar, Benjamin (referee)
The aim of this work was to present different approaches to the proof of Brouwer fixed point theorem and to avoid proofs based on homotopy theory, degree of mapping or any non-trivial algebraic topology. The proofs were chosen so that only a basic knowledge of combinatorics and mathematical analysis is required to understand them and the reader could learn about other fundamental topological theorems. At first, we prove by a combinatorial procedure Borsuk-Ulam theorem from which Brouwer theorem simply follows. We then use the basics of mathematical analysis to prove a theorem known as The hairy ball problem, which also directly implies Brouwer theorem. Finally, we will show an unconventional application of Brouwer theorem to prove the fundamental theorem of algebra. 1
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Gromov-Hausdorff metric
Horský, Miroslav ; Vejnar, Benjamin (advisor) ; Vlasák, Václav (referee)
In this bachelor's thesis we define the notion of Hausdorff metric and Gromov-Hausdorff metric. We will define both metrics in several different ways and then we will show that these definitions are equivalent. We will also proof some fundamental properties of those metrics. At last we will explore Gromov-Hausdorff convergence and proof some strong properties of this convergence. 1
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Dynamical properties of continua
Karasová, Klára ; Vejnar, Benjamin (advisor) ; Bobok, Jozef (referee)
This thesis investigates long-term topological behaviour of continuous self-maps or sets of continuous self-maps of metric spaces, mostly Peano continua. The first chapter is preparatory for the following two and summarize some properties of compact spaces with emphasis on Peano continua. In the second chapter, we give an overview of chaotic features and then we prove that for every Peano continuum X there exists a LEO self- map of X with a dense set of periodic points. In particular, such f is chaotic with respect to widely accepted Devaney' definition of chaos. The third chapter deals with topolog- ical fractals, we prove there a new sufficient condition under which a Peano space is a topological fractal, namely that any Peano continuum with uncountably many local cut- points is a topological fractal. We use this result to partially answer problems concerning regenerating fractals. 1
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H-compactifications of topological spaces
Tížková, Tereza ; Vejnar, Benjamin (advisor) ; Hušek, Miroslav (referee)
H-compactifications form an important type of compactifications, carrying the ex- tra property that all automorphisms of a given topological space can be continuously extended over such compactifications. Van Douwen proved there are only three H-compactifications of the real line and only one of the rationals. Vejnar proved that there are precisely two H-compactifications of higher dimensional Euclidean spaces. The concept of H-compactifications is introduced at the beginning, extra emphasis being put on the Alexandroff and Stone-Čech compacti- fication. We summarize findings that exist about H-compactifications of some well-known spaces. The result we come with in the Chapter 3 is that there is only one H-compactification of the set of all rational sequences, which is precisely the Stone-Čech compactification. The third chapter describes various properties of the set of all rational sequences and its clopen subsets. Some of them - mainly strong zero-dimensionality and strong homogeneity - are then used to reach the said result. In the final Chapter 4, we ask a question about the set of all H-compactifications of the Hilbert space of all square summable real sequences and propose three ways to tackle this problem. We show that under certain conditions, any H-compactification of a space is homeomorphic to...
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Extensions of functions from subspaces of metric spaces
Hevessy, Michal ; Hušek, Miroslav (advisor) ; Vejnar, Benjamin (referee)
Function extension is a classical problem in mathematics. In this thesis we look into an extesion of realvalued functions defined on metric spaces. The first chapter is intro- ductory and describes extension problem. In the second one we discuss a known method for extension of special family of uniformly continuous functions and show that the me- thod can be modified for continuous functions. The third chapter examines a method for extension of continuous functions described by Whitney. Finally, in the last chapter we show a characterisation of uniformly continuous function, having uniformly continuous extensions. 1
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H-compactifications of topological spaces
Tížková, Tereza ; Vejnar, Benjamin (advisor) ; Hušek, Miroslav (referee)
H-compactifications form an important type of compactifications, carrying the ex- tra property that all automorphisms of a given topological space can be continuously extended over such compactifications. Van Douwen proved there are only three H-compactifications of the real line and only one of the rationals. Vejnar proved that there are precisely two H-compactifications of higher dimensional Euclidean spaces. The result we come with in the Chapter 2 is that there is only one H-compactification of the set of all rational sequences, which is precisely the Stone-Čech compactification. For the proof, we use strong zero-dimensionality, strong homogeneity and other properties of the set of all rational sequences and its clopen subsets. In the Chapter 3, we ask an ambitious question about the set of all H-compactifications of the Hilbert space of all square summable real sequences and propose some ways to tackle this problem, e.g. characterizations of the Stone-Čech compactification or tools used to describe H-compactifications of the real space of dimension 2. In the final chapter, we analyze the set of all H-compactifications of a space using a category-theoretic approach and study properties of categories of H-compactifications and functors in such categories. 1
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Homogeneity of topological structures
Vejnar, Benjamin ; Hušek, Miroslav (advisor) ; Pyrih, Pavel (referee)
In the present work we study those compacti cations such that every autohomeomorphism of the base space can be continuously extended over the compacti cation. These are called H-compacti cations. We characterize them by several equivalent conditions and we prove that H-compacti cations of a given space form a complete upper semilattice which is a complete lattice when the given space is supposed to be locally compact. Next, we describe all H-compacti cations of discrete spaces as well as of countable locally compact spaces. It is shown that the only H-compacti cations of Euclidean spaces of dimension at least two are one-point compacti cation and the Cech-Stone compacti cation. Further we get that there are exactly 11 H-compacti cations of a countable sum of Euclidean spaces of dimension at least two and that there are exactly 26 H-compacti cations of a countable sum of real lines. These are all described and a Hasse diagram of a lattice they form is given.
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Abelian regular rings
Vejnar, Benjamin ; Žemlička, Jan (advisor) ; Růžička, Pavel (referee)
Na/ev praco: Abelovsky regularni okruhy Autor: Benjamin Vejnar Katedra (listav): Katedra algebry VcdoLici bakalafske prace: Mgr. Jan 2emlicka, Ph.D. E-mail vedouctho: Jan.Zcmlicka&mJJ. cuni.cz Abstrakt: V pfcdloxene praci studujeme aritmeticke a strukturni vlastnosti abelovsky regularnich okruhu, tedy okruhu, jcjichx ka/xly levy i pravy konecne generova.ny ideal jo generovan idempotentnim prvkem, klery Ic/i v centra danoho okruhu. Napfiklad ka/,dy Boohmv okruh je abelovsky regularni. Venujume ye podininkam, ktere uplne diarakterizuji tn'du abelovsky regu- larnieli okruhu, jako napfiklad silna regularita. Vsimame si souvislosti mexi Booleovou algebrou vsch centralnich idempo1,entu daneho okruhu a hlavnimi idealy. Dale popiHUJeme topologit na nmo/ine visecli prvoidealu a avcdoniujeine si, '/e splyva s Lo])ologii ultrafiltrii na Booleove algebre idciiipotontd. Klicova slova: okruhy, idempoteiitni prvky, silne regularni okruhy Title: Abelian regular rings Author: Benjamin Vejnar Department: Department of Algebra Supervisor: Mgr. Jan Zemlieka, Ph.D. Supervisor's e-mail address: Jan.Zc:ttilicka((})'niff.cu'iii.cz Abstract: In the present work we study arithmetic and structural properties of abelian regular' rings. This means rings whose every left and right finitely generated ideal is generated by an idempotent...
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