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Mountain climbing theorem
Šmídová, Kristýna ; Vejnar, Benjamin (advisor) ; Vlasák, Václav (referee)
Title: Mountain climbing theorem Author: Kristýna Šmídová Department: Department of Mathematical Analysis Supervisor: Mgr. Benjamin Vejnar, Ph.D., Department of Mathematical Ana- lysis Abstract: The subject of this theses is the so-called Muntain Climbers' Pro- blem. We ask for which pairs of continuous functions f, g : [0,1] → [0,1] such that f(0) = g(0) = 0 and f(1) = g(1) = 1 there exist some functions k, h with the same properties such that f (k(x)) = g (h(x)) for all x in the inter- val of [0,1]. For piecewise injective functions we prove the existence using a convenient graph model and handshaking lemma. For locally non-constant functions we provide a constructive proof using uniform convergence. There is also an example of pair of continuons functions for which there exists no suitable pair of functions that solve the problem. The aim is to provide a clear and visual explanation of all the mathematical constructions included. Keywords: continuous function, mountain climber, uniform convergence, hand- shaking lemma
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Topological entropy
Češík, Antonín ; Vejnar, Benjamin (advisor) ; Pražák, Dalibor (referee)
In this thesis we study topological entropy as an invariant of topological dynamical systems. The first chapter contains basic definitions and examples of topological dynamical systems. In the second chapter we introduce the definition of topological entropy on a compact metric space. We study its properties, in particular the fact that it is invariant under conjugacy. The chapter concludes with calculation of topological entropy for the examples introduced in the first chapter. The last chapter deals with generalizing the notion of topological entropy to noncompact metric spaces. The case of piecewise affine maps on the real line is studied in more detail.
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Jordan Curve Theorem
Dudák, Jan ; Vejnar, Benjamin (advisor) ; Kurka, Ondřej (referee)
Title: The Jordan Curve Theorem Author: Jan Dudák Department: Department of Mathematical Analysis Supervisor: Mgr. Benjamin Vejnar, Ph.D., Department of Mathematical Analysis Abstract: The crucial part of this work is the proof of the Jordan curve theorem. To this end, the work starts by introducing necessary notions (e.g. a curve or an arc) and showing their basic properties. Further on, the Brouwer fixed point theorem is proved (in the 2-dimensional case) as well as some of its corollaries which are then used (together with several other proven assertions) in the proof of the Jordan curve theorem. The last chapter of this work briefly informs about possible generalisations of the Jordan curve theorem, referencing to appropriate bibliography. Keywords: Jordan curve, arc, plane, connected component 1
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Universal metric spaces
Raška, Martin ; Hušek, Miroslav (advisor) ; Vejnar, Benjamin (referee)
The thesis covers the properties of isometric embeddings of metric spaces into the Urysohn universal space U (P.S. Urysohn, 1927) and its generalizations (M. Katětov, 1988). The examination of various metric properties of the space U leads to the question of extendability of the embedding ϕ: M → U from a subspace M of a space P onto an embedding Φ: P → U. We approach to this question in situation P = M ∪ {p} in finer form. If ϕ denotes an embedding M → U, let Rϕ denotes the set of images of the point p in U under all possible isometric extensions of the embedding ϕ (we call Rϕ the space of realizations). The main objective of this thesis is answering the following question: Which forms do the spaces Rϕ assume, if ϕ passes all embeddings of the space M into the space U? Corollary 1 and theorem 3 in the II. part of the thesis metrically characterize the family {Rϕ|ϕ: M → U}. We use previous results in part III in order to determine the number of classes of metrically equivalent embeddings of the space M into the space U. As a consequence, we obtain the result of J. Melleray about the homogeneity of the space U.
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Three lakes problem
Šulc, Dominik ; Hencl, Stanislav (advisor) ; Vejnar, Benjamin (referee)
Cílem této práce je nalezení řešení problému tří jezer a podrobný d·kaz jeho správnosti. Problém tří jezer (Lakes of Wada) je úloha, která spočívá v sestrojení tří otevřených souvislých množin v rovině, které se neprotínají a mají společnou hranici. Ukážeme, že takové množiny existují a že kromě uvedených vlastností mohou být dokonce obloukově souvislé. 1
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Big families of incomparable continua
Doležalová, Anna ; Vejnar, Benjamin (advisor) ; Kurka, Ondřej (referee)
The goal of the thesis is to define the basic concepts of continuum theory and explore properties of some special continuous mappings between them. These are used for the construction of infinite families of continua which are incomparable by homeomorphic, open or monotone mappings. Special concern is given to families of dendrites. In particular, we describe an uncountable family of homeomorphically incomparable dendrites, an uncountable family of open incomparable dendrites and a countable family of monotone incomparable local dendrites. Existence of an uncountable family of monotone incomparable dendrites is open problem, in this thesis we describe a family of such dendrites of arbitrary finite cardinality. Powered by TCPDF (www.tcpdf.org)
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Homeomorphisms in topological structures
Vejnar, Benjamin ; Pyrih, Pavel (advisor) ; Charatonik, Włodzimierz (referee) ; Illanes, Alejandro (referee)
In this thesis, we present solutions to several problems concerning one-dimensi- onal continua. We give an inductive description of graphs with a given disconnec- tion number, this answers a question of S. B. Nadler. Further, we state a topo- logical characterization of the Sierpi'nski triangle. In the study of shore sets in dendroids and λ-dendroids we obtain several positive results and we also provide some counterexamples. By doing this, we continue in the recent work of several authors. We are also dealing with the notion of 1 2 -homogeneity and we prove that up to homeomorphism there are only two 1 2 -homogeneous chainable continua with just two end points. We present also a new elegant proof of a classical theorem of Waraszkiewicz. 1
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Connected compactifications
Vaváčková, Martina ; Simon, Petr (advisor) ; Vejnar, Benjamin (referee)
Title: Connected compactifications Author: Martina Vaváčková Department: Department of Theoretical Computer Science and Mathematical Logic Supervisor: prof. RNDr. Petr Simon, DrSc., Department of Theoretical Computer Science and Mathematical Logic Abstract: This thesis deals with connected compactifications of specific Tychonoff spaces. In particular, we are interested in the maximal elements with respect to the partial order over the set of all connected compactifications of a space. First we characterize maximal connected compactifications of spaces containing only finitely many components. We mention examples of spaces which have no connected compactification. Further we study connected compactifications of the rational numbers. We give a construction of a compactification analogical to the construction of the Čech-Stone compactification and we show a necessary and sufficient condition for its connectedness and maximality. Keywords: connected space, compact space, connectification, compactification
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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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