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Ultrafilters and their applications
 Hýlová, Lenka ; Pražák, Dalibor (advisor) ; Vejnar, Benjamin (referee)
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
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