Compact spaces and convergent sequences 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 measure � de�ned on a compact 1 space is sepa rable if the space L ( � ) is sepa ra- ble.
De�nition A measure � de�ned on a compact 1 space is sepa rable if the space L ( � ) is sepa ra- ble. A measure � j � is sepa rable if there is a count- able C � � such that inf f � ( E 4 C ) : C 2 C g = 0 fo r every E 2 �.
De�nition A measure � de�ned on a compact 1 space is sepa rable if the space L ( � ) is sepa ra- ble. A measure � j � is sepa rable if there is a count- able C � � such that inf f � ( E 4 C ) : C 2 C g = 0 fo r every E 2 �. Theo rem (PBN) Minimally generated com- pact spaces admit only sepa rable measures.
De�nition A measure � de�ned on a compact 1 space is sepa rable if the space L ( � ) is sepa ra- ble. A measure � j � is sepa rable if there is a count- able C � � such that inf f � ( E 4 C ) : C 2 C g = 0 fo r every E 2 �. Theo rem (PBN) Minimally generated com- pact spaces admit only sepa rable measures. Mo re p recisely: If � is a measure on a minimally generated compact space, then it is a count- able sum of w eakly unifo rmly regula r measures.
� The algeb ra F in � P ( ! ) is minimally gen- erated;
� The algeb ra F in � P ( ! ) is minimally gen- erated; � One can extend it minimally b y an in�nite subset of ! ;
� The algeb ra F in � P ( ! ) is minimally gen- erated; � One can extend it minimally b y an in�nite subset of ! ; � Consider a minimally generated Bo olean al- geb ra F in � A � P ( ! ) ; such that A cannot b e extended minimally b y a subset of ! .
If K = S tone ( A ), then K do es not contain a nontrivial convergent sequence of natural num- b ers.
If K = S tone ( A ), then K do es not contain a nontrivial convergent sequence of natural num- b ers. Theo rem (Ha ydon) There is a compact but not sequentially compact space ca rrying only sepa rable measures.
If K = S tone ( A ), then K do es not contain a nontrivial convergent sequence of natural num- b ers.
If K = S tone ( A ), then K do es not contain a nontrivial convergent sequence of natural num- b ers. E�mov Problem Is there an in�nite compact space X such that X do es not contain nontriv- ial convergent sequences and do es not contain a cop y of � ! ?
If K = S tone ( A ), then K do es not contain a nontrivial convergent sequence of natural num- b ers. E�mov Problem Is there an in�nite compact space X such that X do es not contain nontriv- ial convergent sequences and do es not contain a cop y of � ! ? If M A is assumed, then there is a (minimally generated) compact space without a cop y of � ! such that convergent sequences consist only of p oint of cha racter c ;
If K = S tone ( A ), then K do es not contain a nontrivial convergent sequence of natural num- b ers.
Is there a minimally generated compacti�ca- tion X of ! such that: � X do es not contain a nontrivial convergent sequence of natural numb ers; � fo r every x 2 X there is a sequence ( n ) k k 2 ! of natural numb ers such that 1 � = lim ( � + ::: + � )? x n n 1 k k !1 k
De�nition A Banach space E has the Mazur �� �� p rop ert y if every x 2 E which is w eak se- � quentially continuous on E b elongs to E . De�nition A Banach space E is said to have the Gelfand{Phillips p rop ert y if every limited subset of E is relatively no rm compact. A subset A of E is limited if � lim sup x ( x ) = 0 ; n n !1 x 2 A � � � fo r every w eak null sequence x in E . n
De�nition A Banach space E has the Mazur �� �� p rop ert y if every x 2 E which is w eak se- � quentially continuous on E b elongs to E . De�nition A Banach space E is said to have the Gelfand{Phillips p rop ert y if every limited subset of E is relatively no rm compact. � there is a Banach space E with the Gelfand{ Philips p rop ert y and without the Mazur p rop- ery;
De�nition A Banach space E has the Mazur �� �� p rop ert y if every x 2 E which is w eak se- � quentially continuous on E b elongs to E . De�nition A Banach space E is said to have the Gelfand{Phillips p rop ert y if every limited subset of E is relatively no rm compact. � there is a Banach space E with the Gelfand{ Philips p rop ert y and without the Mazur p rop- ery; � it is not kno wn if the Mazur p rop ert y im- plies the Gelfand-Phillips p rop ert y .
De�nition A Banach space E has the Mazur �� �� p rop ert y if every x 2 E which is w eak se- � quentially continuous on E b elongs to E . De�nition A Banach space E is said to have the Gelfand{Phillips p rop ert y if every limited subset of E is relatively no rm compact. � there is a Banach space E with the Gelfand{ Philips p rop ert y and without the Mazur p rop- ery; � it is not kno wn if the Mazur p rop ert y im- plies the Gelfand-Phillips p rop ert y . � C ( X ) is an example of Banach space with the Mazur p rop ert y and without the Gelfand- Phillips p rop ert y .
De�nition A Banach space E has the Mazur �� �� p rop ert y if every x 2 E which is w eak se- � quentially continuous on E b elongs to E . De�nition A Banach space E is said to have the Gelfand{Phillips p rop ert y if every limited subset of E is relatively no rm compact. � there is a Banach space E with the Gelfand{ Philips p rop ert y and without the Mazur p rop- ery; � it is not kno wn if the Mazur p rop ert y im- plies the Gelfand-Phillips p rop ert y . � C ( X ) is an example of Banach space with the Mazur p rop ert y and without the Gelfand- Phillips p rop ert y (if only such X exists).
� fo r A � ! de�ne the asymptotic densit y function b y j A \ n j d ( A ) = lim ; n !1 n p rovided this limit exists.
� fo r A � ! de�ne the asymptotic densit y function b y j A \ n j d ( A ) = lim ; n !1 n p rovided this limit exists. � fo r an in�nite M = f m < m < ::: g � ! 1 2 and A � ! de�ne the relative densit y b y d ( A ) = d f k : m 2 A g : M k
� fo r A � ! de�ne the asymptotic densit y function b y j A \ n j d ( A ) = lim ; n !1 n p rovided this limit exists. � fo r an in�nite M = f m < m < ::: g � ! 1 2 and A � ! de�ne the relative densit y b y d ( A ) = d f k : m 2 A g : M k � w e sa y that M is a condeser of a �lter F on ! if d ( F ) = 1 fo r every F 2 F . M
� M is a pseudo{intersection of a �lter F on � ! if M � F fo r every F 2 F . p = min fjAj : A � P ( ! ) generates a �lter without a pseudo{intersection g :
� M is a pseudo{intersection of a �lter F on � ! if M � F fo r every F 2 F . p = min fjAj : A � P ( ! ) generates a �lter without a pseudo{intersection g : � M is a condeser of a �lter F on ! if d ( F ) = M 1 fo r every F 2 F . k = min fjAj : A � P ( ! ) generates a �lter without a condenser g :
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