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Complete Boolean Algebras and Extremally Disconnected Compact Spaces
Starý, Jan ; Simon, Petr (advisor) ; Bukovský, Lev (referee) ; Thümmel, Egbert (referee)
We study the existence of special points in extremally disconnected compact topological spaces that witness their nonhomogeneity. Via Stone duality, we are looking for ultrafilters on complete Boolean algebras with special combinatorial properties. We introduce the notion of a coherent ultrafilter (coherent P-point, coherently selective). We show that generic existence of such ultrafilters on every complete ccc Boolean algebra of weight not exceeding the continuum is consistent with set theory, and that they witness the nonhomogeneity of the corresponding Stone spaces. We study the properties of the order-sequential property on σ-complete Boolean algebras and its relation to measure-theoretic properties. We ask whether the order-sequential topology can be compact in a nontrivial case, and partially answer the question in a special case of the Suslin algebra associated with a Suslin tree.
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Complete Boolean Algebras and Extremally Disconnected Compact Spaces
Starý, Jan ; Simon, Petr (advisor) ; Bukovský, Lev (referee) ; Thümmel, Egbert (referee)
We study the existence of special points in extremally disconnected compact topological spaces that witness their nonhomogeneity. Via Stone duality, we are looking for ultrafilters on complete Boolean algebras with special combinatorial properties. We introduce the notion of a coherent ultrafilter (coherent P-point, coherently selective). We show that generic existence of such ultrafilters on every complete ccc Boolean algebra of weight not exceeding the continuum is consistent with set theory, and that they witness the nonhomogeneity of the corresponding Stone spaces. We study the properties of the order-sequential property on σ-complete Boolean algebras and its relation to measure-theoretic properties. We ask whether the order-sequential topology can be compact in a nontrivial case, and partially answer the question in a special case of the Suslin algebra associated with a Suslin tree.
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Ultrafilters and independent systems
Verner, Jonathan ; Simon, Petr (advisor) ; Zapletal, Jindřich (referee) ; Thümmel, Egbert (referee)
This work presents an overview of several different methods for construct- ing ultrafilters. The first part contains constructions not needing additional assumptions beyond the usual axioms of Set Theory. K. Kunen's method using independent systems for constructing weak P-points is presented. This is followed by a presentation of its application in topology (the proof of the existence of sixteen topological types due to J. van Mill). Finally a new con- struction due to the author is presented together with a proof of his result, the existence of a seventeenth topological type: ω∗ contains a point which is discretely untouchable, is a limit point of a countable set and the countable sets having it as its limit point form a filter. The second part looks at constructions which use additional combina- torial axioms and/or forcing. J. Ketonen's construction of a P-point and A. R. D. Mathias's construction of a Q-point are presented in the first two sections. The next sections concentrate on strong P-points introduced by C. Laflamme. The first of these contains a proof of a new characterization theorem due jointly to the author, A. Blass and M. Hrušák: An ultrafilter is Canjar if and only if it is a strong P-point. A new proof of Canjar's the- orem on the existence of non-dominating filters (Canjar filters) which uses...
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