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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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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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Sobolev mappings and Luzin condition N
Matějka, Milan ; Hencl, Stanislav (advisor) ; Malý, Jan (referee)
A mapping f from R^{n} to R^{n} is said to satisfy the Luzin condition N if f maps sets of measure zero to sets of measure zero. It is known to be valid for mappings in the Sobolev space W^{1,p} for p > n and for p <= n there are counterexamples. The aim of this thesis is to summarize known results and study the validity of Luzin condition N for mappings in the Sobolev space W^{2,p}.
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