|
|
Connected compactifications
Vaváčková, Martina ; Simon, Petr (advisor) ; Vejnar, Benjamin (referee)
Title: Connected compactifications Author: Martina Vaváčková Department: Department of Theoretical Computer Science and Mathematical Logic Supervisor: prof. RNDr. Petr Simon, DrSc., Department of Theoretical Computer Science and Mathematical Logic Abstract: This thesis deals with connected compactifications of specific Tychonoff spaces. In particular, we are interested in the maximal elements with respect to the partial order over the set of all connected compactifications of a space. First we characterize maximal connected compactifications of spaces containing only finitely many components. We mention examples of spaces which have no connected compactification. Further we study connected compactifications of the rational numbers. We give a construction of a compactification analogical to the construction of the Čech-Stone compactification and we show a necessary and sufficient condition for its connectedness and maximality. Keywords: connected space, compact space, connectification, compactification
|
|
| |
|
|
The structure of black hole spacetimes
Haláček, Jakub
In the presented thesis we study taken over solutions of available compactication methods on the Schwarzschild's spacetime and we discuss their properties and analytical structure. Furthermore, we introduce a method of con- struction of coordinates based on analytical requirements placed on the resulting metrics. This procedure is being discussed and applied to Schwarzschild's space- time. Next we apply it to the compactication of the Reissner's Nordström's spacetime and discuss its analytical coverage of the spacetime. Finally, we show a method based on the theory of dierential equations to verify the analytical structure of the metric coecients on the I ± .
|
|
|
Some point-free aspects of connectedness
Jakl, Tomáš ; Pultr, Aleš (advisor)
In this thesis we present the Stone representation theorem, generally known as Stone duality in the point-free context. The proof is choice-free and, since we do not have to be concerned with points, it is by far simpler than the original. For each infinite cardinal κ we show that the counter- part of the κ-complete Boolean algebras is constituted by the κ-basically disconnected Stone frames. We also present a precise characterization of the morphisms which correspond to the κ-complete Boolean homomorphisms. Although Booleanization is not functorial in general, in the part of the dual- ity for extremally disconnected Stone frames it is, and constitutes an equiv- alence of categories. We finish the thesis by focusing on the De Morgan (or extremally disconnected) frames and present a new characterization of these by their superdense sublocales. We also show that in contrast with this phenomenon, a metrizable frame has no non-trivial superdense sublocale; in other words, a non-trivial Čech-Stone compactification of a metrizable frame is never metrizable. 1
|
|
|
Absolute and non-absolute F-Borel spaces
Kovařík, Vojtěch ; Kalenda, Ondřej (advisor)
We investigate F-Borel topological spaces. We focus on finding out how a complexity of a space depends on where the space is embedded. Of a particular interest is the problem of determining whether a complexity of given space X is absolute (that is, the same in every compactification of X). We show that the complexity of metrizable spaces is absolute and provide a sufficient condition for a topological space to be absolutely Fσδ. We then investigate the relation between local and global complexity. To improve our understanding of F-Borel spaces, we introduce different ways of representing an F-Borel set. We use these tools to construct a hierarchy of F-Borel spaces with non-absolute complexity, and to prove several other results. 1
|
|
|
Absolute and non-absolute F-Borel spaces
Kovařík, Vojtěch ; Kalenda, Ondřej (advisor) ; Matheron, Ethienne (referee) ; Holický, Petr (referee)
We investigate F-Borel topological spaces. We focus on finding out how a complexity of a space depends on where the space is embedded. Of a particular interest is the problem of determining whether a complexity of given space X is absolute (that is, the same in every compactification of X). We show that the complexity of metrizable spaces is absolute and provide a sufficient condition for a topological space to be absolutely Fσδ. We then investigate the relation between local and global complexity. To improve our understanding of F-Borel spaces, we introduce different ways of representing an F-Borel set. We use these tools to construct a hierarchy of F-Borel spaces with non-absolute complexity, and to prove several other results. 1
|
|
|
Absolute and non-absolute F-Borel spaces
Kovařík, Vojtěch ; Kalenda, Ondřej (advisor)
We investigate F-Borel topological spaces. We focus on finding out how a complexity of a space depends on where the space is embedded. Of a particular interest is the problem of determining whether a complexity of given space X is absolute (that is, the same in every compactification of X). We show that the complexity of metrizable spaces is absolute and provide a sufficient condition for a topological space to be absolutely Fσδ. We then investigate the relation between local and global complexity. To improve our understanding of F-Borel spaces, we introduce different ways of representing an F-Borel set. We use these tools to construct a hierarchy of F-Borel spaces with non-absolute complexity, and to prove several other results. 1
|
|
|
Some point-free aspects of connectedness
Jakl, Tomáš ; Pultr, Aleš (advisor)
In this thesis we present the Stone representation theorem, generally known as Stone duality in the point-free context. The proof is choice-free and, since we do not have to be concerned with points, it is by far simpler than the original. For each infinite cardinal κ we show that the counter- part of the κ-complete Boolean algebras is constituted by the κ-basically disconnected Stone frames. We also present a precise characterization of the morphisms which correspond to the κ-complete Boolean homomorphisms. Although Booleanization is not functorial in general, in the part of the dual- ity for extremally disconnected Stone frames it is, and constitutes an equiv- alence of categories. We finish the thesis by focusing on the De Morgan (or extremally disconnected) frames and present a new characterization of these by their superdense sublocales. We also show that in contrast with this phenomenon, a metrizable frame has no non-trivial superdense sublocale; in other words, a non-trivial Čech-Stone compactification of a metrizable frame is never metrizable. 1
|
|
| |
|
|
Some point-free aspects of connectedness
Jakl, Tomáš ; Pultr, Aleš (advisor) ; Fiala, Jiří (referee)
In this thesis we present the Stone representation theorem, generally known as Stone duality in the point-free context. The proof is choice-free and, since we do not have to be concerned with points, it is by far simpler than the original. For each infinite cardinal κ we show that the counter- part of the κ-complete Boolean algebras is constituted by the κ-basically disconnected Stone frames. We also present a precise characterization of the morphisms which correspond to the κ-complete Boolean homomorphisms. Although Booleanization is not functorial in general, in the part of the dual- ity for extremally disconnected Stone frames it is, and constitutes an equiv- alence of categories. We finish the thesis by focusing on the De Morgan (or extremally disconnected) frames and present a new characterization of these by their superdense sublocales. We also show that in contrast with this phenomenon, a metrizable frame has no non-trivial superdense sublocale; in other words, a non-trivial Čech-Stone compactification of a metrizable frame is never metrizable. 1
|
guest ::
