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

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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This thesis studies ultrafilters and their various applications in topology, social choice theory and construction of a nonstandard universe. First of all, we introduce basic properties of ultrafilters and show how to use them to construct nonstandard framework. Next, we prove Arrow's impossibility theorem which states that every electoral system with a finite set of voters satisfying certain natural conditions necessarily admits at least one dictator who determines the society's preferences. However, if the set of voters is infinite, this is not true anymore and ultrafilters play a key role in the proof. We present two counterexamples in the infinite case using nonstandard framework. A similar theorem holds in the case where the preferences are real functions. Again, we show two examples of electoral systems that are not dictatorial - one using Banach limits and the other using hyperfinite sums. Finally, we use the ultrafilters to construct the Čech-Stone compactification of natural numbers. We show that the nonstandard enlargement of natural numbers equipped with suitable topology is the Čech-Stone compactification of the set of natural numbers. 1