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Separable reduction theorems, systems of projections and retractions
Cúth, Marek ; Kalenda, Ondřej (advisor) ; Kubiš, Wieslaw (referee) ; Spurný, Jiří (referee)
This thesis consists of four research papers. In the first paper we study whether certain properties of sets (functions) are separably determined. In our results we use the "method of elementary submodels". In the second paper we generalize some results concerning Valdivia compacta (equivalently spaces with a commutative retractional skeleton) to the context of spaces with a retractional skeleton (not necessarily commutative). The third paper further studies the structure of spaces with a projectional (resp. retractional) skeleton. Under certain conditions we prove the existence of a "simultaneous projectional skeleton" and we use this result to prove other statements concerning the structure of spaces with a projectional (resp. retractional) skeleton. In the last paper we study the method of elementary submodels in a greater detail and we compare it with the "method of rich families". 1
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Rich Families of Projections and Retractions
Somaglia, Jacopo ; Kalenda, Ondřej (advisor) ; Aviles, Antonio (referee) ; Kubiš, Wieslaw (referee)
Title: Rich Families of Projections and Retractions Author: Jacopo Somaglia Department: Department of Mathematical Analysis MFF UK (Prague), Department of Mathematics Università degli Studi di Milano (Milan) Supervisors: Prof. RNDr. Ondřej Kalenda PhD DSc. Department of Mathematical Analysis MFF UK (Prague), Prof. Dr. Clemente Zanco Department of Mathematics Università degli Studi di Milano (Milan) Abstract: We deal with problems on non-separable Banach spaces and non-metrizable compact spaces. In particular these problems concern Banach spaces with a projectional skeleton and compact spaces with a retractional skeleton. A projectional (resp. retractional) skeleton is a family of continuous projections (resp. retractions) on a Banach (resp. compact) space, which satisfies certain compatibility properties. Banach spaces with projectional skeleton and compact spaces with retractional skeleton can be viewed as non-commutative version of Plichko Banach spaces and Valdivia compact spaces respectively. The thesis is split into three chapters. Each chapter consists of a submitted/published paper concerning different problems in this area. In the first chapter, On the class of continuous images of non-commutative Valdivia compacta, we investigate the stability of some topological properties in the class of weakly...
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Between homogeneity and rigidity
Grebík, Jan ; Kubiš, Wieslaw (advisor) ; Šaroch, Jan (referee)
Studujeme nespočetné struktury, které splňují exstension property vzhledem k nějaké Fraïssé třídě C. Takovým strukturám říkáme Fraïssé-like struktury. Tyto struktury nejsou většinou jednoznačně určeny. Je známo, že pokud existuje Katětov funktor pro C, pak existují Fraïssé-like struktury libovolné kardinality s bohatou grupou automorfismů. Ukážeme, že v případě třídy všech konečných grafů a všech konečných metrických prostorů existuje Fraïssé-like struktura, která má kardinalitu ℵ1 a její grupa automorfismů je triviální. Dále zodpovíme otázku z W. Kubi's, D. Mašulovi'c, Katětov functors, to appear in Applied Categorical Structures tak, že nalezneme Fraïssé třídu bez Katětova funktoru. 1
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Separable reduction theorems, systems of projections and retractions
Cúth, Marek ; Kalenda, Ondřej (advisor) ; Kubiš, Wieslaw (referee) ; Spurný, Jiří (referee)
This thesis consists of four research papers. In the first paper we study whether certain properties of sets (functions) are separably determined. In our results we use the "method of elementary submodels". In the second paper we generalize some results concerning Valdivia compacta (equivalently spaces with a commutative retractional skeleton) to the context of spaces with a retractional skeleton (not necessarily commutative). The third paper further studies the structure of spaces with a projectional (resp. retractional) skeleton. Under certain conditions we prove the existence of a "simultaneous projectional skeleton" and we use this result to prove other statements concerning the structure of spaces with a projectional (resp. retractional) skeleton. In the last paper we study the method of elementary submodels in a greater detail and we compare it with the "method of rich families". 1
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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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