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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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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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Forcing, deskriptivní teorie množin, analýza
Doucha, Michal ; Zapletal, Jindřich (advisor) ; Zelený, Miroslav (referee) ; Kubiš, Wieslaw (referee)
The dissertation thesis consists of two thematic parts. The first part, i.e. chapters 2, 3 and 4, contains results concerning the topic of a new book of the supervisor and coauthors V. Kanovei and M. Sabok "Canonical Ramsey Theory on Polish Spaces". In Chapter 2, there is proved a canonization of all equivalence relations Borel reducible to equivalences definable by analytic P-ideals for the Silver ideal. Moreover, it investigates and classifies sube- quivalences of the equivalence relation E0. In Chapter 3, there is proved a canonization of all equivalence relations Borel reducible to equivalences de- finable by Fσ P-ideals for the Laver ideal and in Chapter 4, we prove the canonization for all analytic equivalence relations for the ideal derived from the Carlson-Simpson (Dual Ramsey) theorem. The second part, consisting of Chapter 5, deals with the existence of universal and ultrahomogeneous Polish metric structures. For instance, we construct a universal Polish metric space which is moreover equipped with countably many closed relations or with a Lipschitz function to an arbitrarily chosen Polish metric space. This work can be considered as an extension of the result of P. Urysohn who constructed a universal and ultrahomogeneous Polish metric space.
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