Compact spaces and convergent sequences Piotr Bo ro - - PDF document

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Compact spaces and convergent sequences Piotr Bo ro - - PDF document

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Compact spaces and convergent sequences Piotr Bo ro dulin-Nadzieja Inst ytut Matemat yczny Uniw ersytetu W ro c la wskiego Denition A Bo olean algeb ra B extends an algeb ra A minimally if A B and there

slide-1
SLIDE 1 Compact spaces and convergent sequences Piotr Bo ro dulin-Nadzieja Inst ytut Matemat yczny Uniw ersytetu W ro c la wskiego
slide-2
SLIDE 2 Denition A Bo
  • lean
algeb ra B extends an algeb ra A minimally if A
  • B
and there is no algeb ra C such that A ( C ( B .
slide-3
SLIDE 3 Denition A Bo
  • lean
algeb ra B extends an algeb ra A minimally if A
  • B
and there is no algeb ra C such that A ( C ( B . Denition A Bo
  • lean
algeb ra A is minimally generated if there is a sequence (A
  • )
< such that
  • A
= f0 ; 1 g;
  • A
+1 extends A
  • minimally
fo r
  • <
;
  • A
  • =
S < A
  • fo
r limit
  • ;
  • A
= S < A
  • .
slide-4
SLIDE 4 Denition A Bo
  • lean
algeb ra A is minimally generated if there is a sequence (A
  • )
< such that
  • A
= f0 ; 1 g;
  • A
+1 extends A
  • minimally
fo r
  • <
;
  • A
  • =
S < A
  • fo
r limit
  • ;
  • A
= S < A
  • .
A top
  • logical
space K is minimally generated if it is a Stone space
  • f
a minimally generated Bo
  • lean
algeb ra.
slide-5
SLIDE 5 Denition A measure
  • dened
  • n
a compact space is sepa rable if the space L 1 ( ) is sepa ra- ble.
slide-6
SLIDE 6 Denition A measure
  • dened
  • n
a compact space is sepa rable if the space L 1 ( ) 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 = fo r every E 2 .
slide-7
SLIDE 7 Denition A measure
  • dened
  • n
a compact space is sepa rable if the space L 1 ( ) 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 = fo r every E 2 . Theo rem (PBN) Minimally generated com- pact spaces admit
  • nly
sepa rable measures.
slide-8
SLIDE 8 Denition A measure
  • dened
  • n
a compact space is sepa rable if the space L 1 ( ) 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 = fo r every E 2 . Theo rem (PBN) Minimally generated com- pact spaces admit
  • nly
sepa rable measures. Mo re p recisely: If
  • is
a measure
  • n
a minimally generated compact space, then it is a count- able sum
  • f
w eakly unifo rmly regula r measures.
slide-9
SLIDE 9
  • The
algeb ra F in
  • P
(! ) is minimally gen- erated;
slide-10
SLIDE 10
  • The
algeb ra F in
  • P
(! ) is minimally gen- erated;
  • One
can extend it minimally b y an innite subset
  • f
! ;
slide-11
SLIDE 11
  • The
algeb ra F in
  • P
(! ) is minimally gen- erated;
  • One
can extend it minimally b y an innite subset
  • f
! ;
  • Consider
a minimally generated Bo
  • lean
al- geb ra F in
  • A
  • P
(! ); such that A cannot b e extended minimally b y a subset
  • f
! .
slide-12
SLIDE 12 If K = S tone (A ), then K do es not contain a nontrivial convergent sequence
  • f
natural num- b ers.
slide-13
SLIDE 13 If K = S tone (A ), then K do es not contain a nontrivial convergent sequence
  • f
natural num- b ers. Theo rem (Ha ydon) There is a compact but not sequentially compact space ca rrying
  • nly
sepa rable measures.
slide-14
SLIDE 14 If K = S tone (A ), then K do es not contain a nontrivial convergent sequence
  • f
natural num- b ers.
slide-15
SLIDE 15 If K = S tone (A ), then K do es not contain a nontrivial convergent sequence
  • f
natural num- b ers. Emov Problem Is there an innite compact space X such that X do es not contain nontriv- ial convergent sequences and do es not contain a cop y
  • f
  • !
?
slide-16
SLIDE 16 If K = S tone (A ), then K do es not contain a nontrivial convergent sequence
  • f
natural num- b ers. Emov Problem Is there an innite compact space X such that X do es not contain nontriv- ial convergent sequences and do es not contain a cop y
  • f
  • !
? If M A is assumed, then there is a (minimally generated) compact space without a cop y
  • f
  • !
such that convergent sequences consist
  • nly
  • f
p
  • int
  • f
cha racter c;
slide-17
SLIDE 17 If K = S tone (A ), then K do es not contain a nontrivial convergent sequence
  • f
natural num- b ers.
slide-18
SLIDE 18 Is there a minimally generated compactica- tion X
  • f
! such that:
  • X
do es not contain a nontrivial convergent sequence
  • f
natural numb ers;
  • fo
r every x 2 X there is a sequence (n k ) k 2!
  • f
natural numb ers such that
  • x
= lim k !1 1 k ( n 1 + ::: +
  • n
k )?
slide-19
SLIDE 19 Denition A Banach space E has the Mazur p rop ert y if every x
  • 2
E
  • which
is w eak se- quentially continuous
  • n
E
  • b
elongs to E . Denition A Banach space E is said to have the Gelfand{Phillips p rop ert y if every limited subset
  • f
E is relatively no rm compact. A subset A
  • f
E is limited if lim n!1 sup x2A x
  • n
(x) = 0 ; fo r every w eak
  • null
sequence x
  • n
in E
  • .
slide-20
SLIDE 20 Denition A Banach space E has the Mazur p rop ert y if every x
  • 2
E
  • which
is w eak se- quentially continuous
  • n
E
  • b
elongs to E . Denition A Banach space E is said to have the Gelfand{Phillips p rop ert y if every limited subset
  • f
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;
slide-21
SLIDE 21 Denition A Banach space E has the Mazur p rop ert y if every x
  • 2
E
  • which
is w eak se- quentially continuous
  • n
E
  • b
elongs to E . Denition A Banach space E is said to have the Gelfand{Phillips p rop ert y if every limited subset
  • f
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 .
slide-22
SLIDE 22 Denition A Banach space E has the Mazur p rop ert y if every x
  • 2
E
  • which
is w eak se- quentially continuous
  • n
E
  • b
elongs to E . Denition A Banach space E is said to have the Gelfand{Phillips p rop ert y if every limited subset
  • f
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
  • f
Banach space with the Mazur p rop ert y and without the Gelfand- Phillips p rop ert y .
slide-23
SLIDE 23 Denition A Banach space E has the Mazur p rop ert y if every x
  • 2
E
  • which
is w eak se- quentially continuous
  • n
E
  • b
elongs to E . Denition A Banach space E is said to have the Gelfand{Phillips p rop ert y if every limited subset
  • f
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
  • f
Banach space with the Mazur p rop ert y and without the Gelfand- Phillips p rop ert y (if
  • nly
such X exists).
slide-24
SLIDE 24
  • fo
r A
  • !
dene the asymptotic densit y function b y d(A ) = lim n!1 jA \ n j n ; p rovided this limit exists.
slide-25
SLIDE 25
  • fo
r A
  • !
dene the asymptotic densit y function b y d(A ) = lim n!1 jA \ n j n ; p rovided this limit exists.
  • fo
r an innite M = fm 1 < m 2 < :::g
  • !
and A
  • !
dene the relative densit y b y d M (A ) = dfk : m k 2 A g:
slide-26
SLIDE 26
  • fo
r A
  • !
dene the asymptotic densit y function b y d(A ) = lim n!1 jA \ n j n ; p rovided this limit exists.
  • fo
r an innite M = fm 1 < m 2 < :::g
  • !
and A
  • !
dene the relative densit y b y d M (A ) = dfk : m k 2 A g:
  • w
e sa y that M is a condeser
  • f
a lter F
  • n
! if d M (F ) = 1 fo r every F 2 F .
slide-27
SLIDE 27
  • M
is a pseudo{intersection
  • f
a lter F
  • n
! if M
  • F
fo r every F 2 F . p = minfjAj : A
  • P
(! ) generates a lter without a pseudo{intersection g:
slide-28
SLIDE 28
  • M
is a pseudo{intersection
  • f
a lter F
  • n
! if M
  • F
fo r every F 2 F . p = minfjAj : A
  • P
(! ) generates a lter without a pseudo{intersection g:
  • M
is a condeser
  • f
a lter F
  • n
! if d M (F ) = 1 fo r every F 2 F . k = minfjAj : A
  • P
(! ) generates a lter without a condenser g:
slide-29
SLIDE 29
  • M
is a pseudo{intersection
  • f
a lter F
  • n
! if M
  • F
fo r every F 2 F . p = minfjAj : A
  • P
(! ) generates a lter without a pseudo{intersection g:
  • M
is a condeser
  • f
a lter F
  • n
! if d M (F ) = 1 fo r every F 2 F . k = minfjAj : A
  • P
(! ) generates a lter without a condenser g:
  • it
is easy to see that if M is a pseudo{ intersection
  • f
a lter F , then it is a con- denser
  • f
this lter. Therefo re p
  • k:
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SLIDE 30 If A is a minimally generated Bo
  • lean
algeb ra such that no ultralter
  • n
A has an innite pseudo{intersection but every ultralter has a condenser, then X = S tone (A ) satises the de- manded p rop erties. In pa rticula r, C (X ) has the Mazur p rop ert y but do es not have the Gelfand{Phillips p rop ert y .
slide-31
SLIDE 31 If A is a minimally generated Bo
  • lean
algeb ra such that no ultralter
  • n
A has an innite pseudo{intersection but every ultralter has a condenser, then X = S tone (A ) satises the de- manded p rop erties. In pa rticula r, C (X ) has the Mazur p rop ert y but do es not have the Gelfand{Phillips p rop ert y . C
  • n
(p < k)?
slide-32
SLIDE 32 A lter F
  • n
! is feeble if there a nite{to{one function f : ! ! ! such that f [F ] is co{nite fo r every F 2 F .
slide-33
SLIDE 33 A lter F
  • n
! is feeble if there a nite{to{one function f : ! ! ! such that f [F ] is co{nite fo r every F 2 F . Every lter with a condenser is feeble.
slide-34
SLIDE 34 A lter F
  • n
! is feeble if there a nite{to{one function f : ! ! ! such that f [F ] is co{nite fo r every F 2 F . Every lter with a condenser is feeble. Theo rem If h < b, then there is a minimally generated Bo
  • lean
algeb ra such that no ultra- lter
  • n
A has an innite pseudo{intersection but every ultralter
  • n
A is feeble.