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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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Families of connected spaces
Bartoš, Adam ; Vejnar, Benjamin (advisor) ; Charatonik, Włodzimierz (referee) ; Hušek, Miroslav (referee)
Families of connected spaces Adam Bartoš Abstract We deal with two completely different kinds of connected spaces - maximal connected spaces and metrizable continua. A topologi- cal space is maximal connected if it is connected, but every strictly finer topology on the same base set is disconnected. Here, the name "Families of connected spaces" refers to the collection of all connected topologies on a given set, which is ordered by inclusion, and maximal connected topologies are its maximal elements. We study the con- struction of tree sums of topological spaces, and how this construc- tion preserves maximal connectedness. We also characterize finitely generated maximal connected spaces as T1 2 -compatible tree sums of copies of the Sierpiński space. On the other hand, we are interested in a general question when for a given class of continua there exists a metrizable compactum whose set of components is equivalent to the given class. (Two classes are equivalent if they contain the same spaces up to homeomorphic copies.) We introduce compactifiable, Polishable, strongly compactifiable, and strongly Polishable classes of compacta, and we investigate their properties. This is related to the descriptive complexity of equivalent realizations of the given class in the hyper- space of all compacta. We prove that...
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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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