Filtri i ultrafiltri

U ovom diplomskom radu smo proučavali filtre i ultrafiltre, te njihovu primjenu. Cilj rada je bio dokazati Silverov teorem, tj. generaliziranu hipotezu kontinuuma za singularne kardinale. U poglavlju 1 smo promatrali filtre i ideale, te njihove primjere. Definirali smo pojam glavnog filtra, te maksimalnog filtra. Dokazali smo da je presjek filtera opet filter, te da je unija ideala opet ideal. Dokazali smo da filtri imaju svojstvo konačnih presjeka i da se svaka familija sa tim svojstvom može proširiti do filtera. U ovom poglavlju smo definirali i gustoću na skupu prirodnih brojeva. Nakon filtera, u poglavlju 2 proučavali smo ultrafiltere koji su zapravo maksimalni filtri. Svaki filter se može proširiti do ultrafiltra. Pomoću ultrafiltera smo definirali $\mathcal{U}$-limes, pomoću kojeg smo definirali netrivijalnu mjeru na skupu prirodnih brojeva. Osim toga dokazali smo da je $\mathcal{U}$-limes linearan, te da je za konvergenciju niza dovoljno da niz bude ograničen. U poglavlju 3 smo prvo definirali zatvorene neomeđene skupove. Dokazali smo da je prebrojivi presjek zatvorenih neomeđenih skupova ponovo zatvoren i neomeđen. Definirali smo i zatvoreni neomeđeni filter čije proširenje do ultrafiltera smo iskoristili prilikom dokaza Silverovog teorema. Nakon zatvorenih neomeđenih skupova smo definirali stacionarne skupove. Najvažniji rezultat u ovom poglavlju je teorem 3.2.6 koji kaže da je skup S stacionaran ako i samo ako je svaka regresivna funkcija na S konstanta na nekom neomeđenom podskupu od S . Svojstva stacionarnih skupova smo također koristili u dokazu Silverovog teorema. Silverov teorem dokazali smo u poglavlju 4. Prije iskaza teorema smo definirali pojmove regularnih i singularnih kardinalnih brojeva, te pojam kofinalnosti za granični ordinalni broj. Nakon toga smo pomoću niza lema dokazali Silverov teorem.In this graduate thesis we have studied filters and ultrafilters, and their application. The goal of this graduate thesis was to prove the Silver theorem, ie. the generalized hypothesis of continuum for singular cardinals. In chapter 1 we have studied the filters and ideals and their examples. We defined the concept of the principal filter and the maximal filter. We proved that intersection of filters is again filter, and that the union of ideals is ideal again. We proved that filters have a finite intersection property and that any collection with this property can be exdend to some filter. In this chapter we also defined the density on the set of natural numbers. Next, in the chapter 2 we studied ultrafilters which are actually maximal filters. Each filter can be extended to ultrafilter. By using the ultrafilters we have defined $\mathcal{U}$-limes, by which we have defined the nontrivial measure at the set of natural numbers. In addition, we proved that $\mathcal{U}$-limes is linear, and that for the convergence of sequence is sufficient that the array is limited. In chapter 3 first we defined closed unbounded sets. We proved that the intersection of countably many closed unbounded sets is closed unbounded. We have also defined a closed unbounded filter whose extension to the ultrafilter was used in proof of Silver’s theorem. After closed unbounded sets we defined stationary sets. The most significant result in this chapter is the theorem 3.2.6 which states that the set S is stationary if and only if every regressive function at S is constant on an unbounded subset of S . The properties of stationary sets were also used in the proof of Silver’s theorem. Silver’s theorem has been proven in the chapter 4. Before the theorem we have defined the regular and singular cardinal numbers, and the cofinality for the limit ordinal number. Then, by use of several lemmas, we proved Silver’s theorem

Fat subsets of P kappa (lambda)

For a subset of a cardinal greater than ω1, fatness is strictly stronger than stationarity and strictly weaker than being closed unbounded. For many regular cardinals, being fat is a sufficient condition for having a closed unbounded subset in some generic extension. In this work we characterize fatness for subsets of Pκ(λ). We prove that for many regular cardinals κ and λ, a fat subset of Pκ(λ) obtains a closed unbounded subset in a cardinal-preserving generic extension. Additionally, we work out the conflict produced by two different definitions of fat subset of a cardinal, and introduce a novel (not model-theoretic) proof technique for adding a closed unbounded subset to a fat subset of a cardinal

Filtri i ultrafiltri

U ovom diplomskom radu smo proučavali filtre i ultrafiltre, te njihovu primjenu. Cilj rada je bio dokazati Silverov teorem, tj. generaliziranu hipotezu kontinuuma za singularne kardinale. U poglavlju 1 smo promatrali filtre i ideale, te njihove primjere. Definirali smo pojam glavnog filtra, te maksimalnog filtra. Dokazali smo da je presjek filtera opet filter, te da je unija ideala opet ideal. Dokazali smo da filtri imaju svojstvo konačnih presjeka i da se svaka familija sa tim svojstvom može proširiti do filtera. U ovom poglavlju smo definirali i gustoću na skupu prirodnih brojeva. Nakon filtera, u poglavlju 2 proučavali smo ultrafiltere koji su zapravo maksimalni filtri. Svaki filter se može proširiti do ultrafiltra. Pomoću ultrafiltera smo definirali $\mathcal{U}$-limes, pomoću kojeg smo definirali netrivijalnu mjeru na skupu prirodnih brojeva. Osim toga dokazali smo da je $\mathcal{U}$-limes linearan, te da je za konvergenciju niza dovoljno da niz bude ograničen. U poglavlju 3 smo prvo definirali zatvorene neomeđene skupove. Dokazali smo da je prebrojivi presjek zatvorenih neomeđenih skupova ponovo zatvoren i neomeđen. Definirali smo i zatvoreni neomeđeni filter čije proširenje do ultrafiltera smo iskoristili prilikom dokaza Silverovog teorema. Nakon zatvorenih neomeđenih skupova smo definirali stacionarne skupove. Najvažniji rezultat u ovom poglavlju je teorem 3.2.6 koji kaže da je skup S stacionaran ako i samo ako je svaka regresivna funkcija na S konstanta na nekom neomeđenom podskupu od S . Svojstva stacionarnih skupova smo također koristili u dokazu Silverovog teorema. Silverov teorem dokazali smo u poglavlju 4. Prije iskaza teorema smo definirali pojmove regularnih i singularnih kardinalnih brojeva, te pojam kofinalnosti za granični ordinalni broj. Nakon toga smo pomoću niza lema dokazali Silverov teorem.In this graduate thesis we have studied filters and ultrafilters, and their application. The goal of this graduate thesis was to prove the Silver theorem, ie. the generalized hypothesis of continuum for singular cardinals. In chapter 1 we have studied the filters and ideals and their examples. We defined the concept of the principal filter and the maximal filter. We proved that intersection of filters is again filter, and that the union of ideals is ideal again. We proved that filters have a finite intersection property and that any collection with this property can be exdend to some filter. In this chapter we also defined the density on the set of natural numbers. Next, in the chapter 2 we studied ultrafilters which are actually maximal filters. Each filter can be extended to ultrafilter. By using the ultrafilters we have defined $\mathcal{U}$-limes, by which we have defined the nontrivial measure at the set of natural numbers. In addition, we proved that $\mathcal{U}$-limes is linear, and that for the convergence of sequence is sufficient that the array is limited. In chapter 3 first we defined closed unbounded sets. We proved that the intersection of countably many closed unbounded sets is closed unbounded. We have also defined a closed unbounded filter whose extension to the ultrafilter was used in proof of Silver’s theorem. After closed unbounded sets we defined stationary sets. The most significant result in this chapter is the theorem 3.2.6 which states that the set S is stationary if and only if every regressive function at S is constant on an unbounded subset of S . The properties of stationary sets were also used in the proof of Silver’s theorem. Silver’s theorem has been proven in the chapter 4. Before the theorem we have defined the regular and singular cardinal numbers, and the cofinality for the limit ordinal number. Then, by use of several lemmas, we proved Silver’s theorem

Filtri i ultrafiltri

U ovom diplomskom radu smo proučavali filtre i ultrafiltre, te njihovu primjenu. Cilj rada je bio dokazati Silverov teorem, tj. generaliziranu hipotezu kontinuuma za singularne kardinale. U poglavlju 1 smo promatrali filtre i ideale, te njihove primjere. Definirali smo pojam glavnog filtra, te maksimalnog filtra. Dokazali smo da je presjek filtera opet filter, te da je unija ideala opet ideal. Dokazali smo da filtri imaju svojstvo konačnih presjeka i da se svaka familija sa tim svojstvom može proširiti do filtera. U ovom poglavlju smo definirali i gustoću na skupu prirodnih brojeva. Nakon filtera, u poglavlju 2 proučavali smo ultrafiltere koji su zapravo maksimalni filtri. Svaki filter se može proširiti do ultrafiltra. Pomoću ultrafiltera smo definirali $\mathcal{U}$-limes, pomoću kojeg smo definirali netrivijalnu mjeru na skupu prirodnih brojeva. Osim toga dokazali smo da je $\mathcal{U}$-limes linearan, te da je za konvergenciju niza dovoljno da niz bude ograničen. U poglavlju 3 smo prvo definirali zatvorene neomeđene skupove. Dokazali smo da je prebrojivi presjek zatvorenih neomeđenih skupova ponovo zatvoren i neomeđen. Definirali smo i zatvoreni neomeđeni filter čije proširenje do ultrafiltera smo iskoristili prilikom dokaza Silverovog teorema. Nakon zatvorenih neomeđenih skupova smo definirali stacionarne skupove. Najvažniji rezultat u ovom poglavlju je teorem 3.2.6 koji kaže da je skup S stacionaran ako i samo ako je svaka regresivna funkcija na S konstanta na nekom neomeđenom podskupu od S . Svojstva stacionarnih skupova smo također koristili u dokazu Silverovog teorema. Silverov teorem dokazali smo u poglavlju 4. Prije iskaza teorema smo definirali pojmove regularnih i singularnih kardinalnih brojeva, te pojam kofinalnosti za granični ordinalni broj. Nakon toga smo pomoću niza lema dokazali Silverov teorem.In this graduate thesis we have studied filters and ultrafilters, and their application. The goal of this graduate thesis was to prove the Silver theorem, ie. the generalized hypothesis of continuum for singular cardinals. In chapter 1 we have studied the filters and ideals and their examples. We defined the concept of the principal filter and the maximal filter. We proved that intersection of filters is again filter, and that the union of ideals is ideal again. We proved that filters have a finite intersection property and that any collection with this property can be exdend to some filter. In this chapter we also defined the density on the set of natural numbers. Next, in the chapter 2 we studied ultrafilters which are actually maximal filters. Each filter can be extended to ultrafilter. By using the ultrafilters we have defined $\mathcal{U}$-limes, by which we have defined the nontrivial measure at the set of natural numbers. In addition, we proved that $\mathcal{U}$-limes is linear, and that for the convergence of sequence is sufficient that the array is limited. In chapter 3 first we defined closed unbounded sets. We proved that the intersection of countably many closed unbounded sets is closed unbounded. We have also defined a closed unbounded filter whose extension to the ultrafilter was used in proof of Silver’s theorem. After closed unbounded sets we defined stationary sets. The most significant result in this chapter is the theorem 3.2.6 which states that the set S is stationary if and only if every regressive function at S is constant on an unbounded subset of S . The properties of stationary sets were also used in the proof of Silver’s theorem. Silver’s theorem has been proven in the chapter 4. Before the theorem we have defined the regular and singular cardinal numbers, and the cofinality for the limit ordinal number. Then, by use of several lemmas, we proved Silver’s theorem

Topological aspects of poset spaces

We study two classes of spaces whose points are filters on partially ordered sets. Points in MF spaces are maximal filters, while points in UF spaces are unbounded filters. We give a thorough account of the topological properties of these spaces. We obtain a complete characterization of the class of countably based MF spaces: they are precisely the second-countable T_1 spaces with the strong Choquet property. We apply this characterization to domain theory to characterize the class of second-countable spaces with a domain representation.Comment: 29 pages. To be published in the Michigan Mathematical Journa