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Specialni bezbodove prostory
Novák, Jan ; Pultr, Aleš (advisor) ; Klazar, Martin (referee)
1 This thesis concerns separation axioms in point-free topology. We introduce a notion of weak inclusion, which is a relation on a frame that is weaker than the relation ≤. Weak inclusions provide a uniform way to work with standard separation axioms such as subfitness, fitness, and regularity. Proofs using weak inclusions often bring new insight into the nature of the axioms. We focus on results related to the axiom of subfitness. We study a sublocale which is defined as the intersection of all the codense sublocales of a frame. We show that it need not be subfit. For spacial frames, it need not be spacial.
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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
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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
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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
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Separation axioms
Ha, Karel ; Pultr, Aleš (advisor) ; Loebl, Martin (referee)
The classical (point-set) topology concerns points and relationships between points and subsets. Omitting points and considering only the structure of open sets leads to the notion of frames, that is, a complete lattice satisfying the dis- tributive law b ∧ A = {b ∧ a | a ∈ A}, the crucial concept of point-free topology. This pointless approach-while losing hardly any information-provides us with deeper insights on topology. One such example is the study of separation axioms. This thesis focuses on the Ti-axioms (for i = 0, 1, 2, 3, 31 2 , 4): properties of topological spaces which regard the separation of points, points from closed sets, and closed sets from one another. In this text we discuss their point-free counterparts and how they relate to each other. 1
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