Minimally generated Bo olean algeb ras Piotr Bo ro dulin-Nadzieja Inst ytut Matemat yczny Uniw ersytetu W ro c la wskiego
De�nition A Bo olean algeb ra B extends an algeb ra A minimally if A � B and there is no algeb ra C such that A ( C ( B .
De�nition A Bo olean algeb ra B extends an algeb ra A minimally if A � B and there is no algeb ra C such that A ( C ( B . De�nition A Bo olean algeb ra A is minimally generated if there is a sequence ( A ) such � �<� that � A = f 0 ; 1 g ; 0 � A extends A minimally fo r � < � ; � � +1 S � A = A fo r limit � ; � � �<� S � A = A . � �<�
De�nition A Bo olean algeb ra A is minimally generated if there is a sequence ( A ) such � �<� that � A = f 0 ; 1 g ; 0 � A extends A minimally fo r � < � ; � � +1 S � A = A fo r limit � ; � � �<� S � A = A . � �<� A top ological space K is minimally generated if it is a Stone space of a minimally generated Bo olean algeb ra.
De�nition A Bo olean algeb ra A is small if it do es not contain an uncountable indep endent family .
De�nition A Bo olean algeb ra A is small if it do es not contain an uncountable indep endent family . Theo rem (Kopp elb erg) Minimally generated Bo olean algeb ras a re small.
De�nition A Bo olean algeb ra A is small if it do es not contain an uncountable indep endent family . Theo rem (Kopp elb erg) Minimally generated Bo olean algeb ras a re small. Theo rem (Kopp elb erg) There is a small Bo olean algeb ra which is not minimally generated.
F act Every (�nitely additive) measure � on a Bo olean algeb ra A can b e uniquely extended 0 to a Radon measure � on the Stone space of A .
F act Every (�nitely additive) measure � on a Bo olean algeb ra A can b e uniquely extended 0 to a Radon measure � on the Stone space of A . De�nition A measure � de�ned on a Bo olean 0 algeb ra A is sepa rable if L ( � ) is sepa rable. 1
F act Every (�nitely additive) measure � on a Bo olean algeb ra A can b e uniquely extended 0 to a Radon measure � on the Stone space of A . De�nition A measure � de�ned on a Bo olean 0 algeb ra A is sepa rable if L ( � ) is sepa rable. 1 F act Every big Bo olean algeb ra ca rries a non- sepa rable measure.
F act Every (�nitely additive) measure � on a Bo olean algeb ra A can b e uniquely extended 0 to a Radon measure � on the Stone space of A . De�nition A measure � de�ned on a Bo olean 0 algeb ra A is sepa rable if L ( � ) is sepa rable. 1 F act Every big Bo olean algeb ra ca rries a non- sepa rable measure. Theo rem (F remlin) M A ( ! ) implies that small 1 Bo olean algeb ras admit only sepa rable mea- sures.
Theo rem Minimally generated Bo olean alge- b ras admit only sepa rable measures.
Theo rem Minimally generated Bo olean alge- b ras admit only sepa rable measures. Mo re p recisely: If � is a measure on a minimally generated Bo olean algeb ra then it is a count- able sum of w eakly unifo rmly regula r measures.
Co rolla ry: The follo wing classes of Bo olean algeb ras (spaces) admit only sepa rable measures: � interval algeb ras (o rdered spaces); � tree algeb ras; � sup eratomic algeb ras (scattered spaces); � monotonically no rmal spaces.
Co rolla ry: The follo wing classes of Bo olean algeb ras (spaces) a re not included (at least, not in ZF C) in the class of minimally generated Bo olean algeb ras: � retractive algeb ras; � Co rson compacta.
Another co rolla ries: A length of minimally generated Bo olean alge- b ra has to b e a limit o rdinal but not necessa rily a ca rdinal. In the realm of minimally generated spaces ccc = sepa rabilit y Mo reover, if w e assume Suslin Conjecture then ccc = existence of strictly p ositive measure
E�mov Problem Is there a compact in�nite space K such that � K do es not contain a nontrivial convergent sequence; � K do es not contain a cop y of � ! ?
E�mov Problem Is there a compact in�nite space K such that � K do es not contain a nontrivial convergent sequence; � K do es not contain a cop y of � ! ? Theo rem (F edo rchuk) CH implies the exis- tence of such space. It is not kno wn if it is consistent to assume that there is no E�mov space.
Problem Ho w to cha racterize minimally gen- erated spaces? Theo rem (Kopp elb erg) Every minimally gen- erated space has a tree � -base.
Problem Ho w to cha racterize minimally gen- erated spaces? Theo rem (Kopp elb erg) Every minimally gen- erated space has a tree � -base. Question Is it true that every minimally gen- erated space is discretely generated?
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