CALIBERS OF COMPACT SPACES1 · CALIBERS OF COMPACT SPACES1 BY S. ARGYROS AND A. TSARPALIAS...

TRANSACTIONS OF THEAMERICAN MATHEMATICAL SOCIETYVolume 270, Number I, March 1982

CALIBERS OF COMPACT SPACES1

BY

S. ARGYROS AND A. TSARPALIAS

Abstract. Let A' be a compact Hausdorff space and k its Souslin number.2 We

prove that if a is a cardinal such that either a and cf(a) are greater than k and

strongly «-inaccessible or else a is regular and greater than k, then X has (a,%a)

caliber.

Restricting our interest to the category of compact spaces X with S{ X) = u+ (i.e.

X satisfy the countable chain condition), the above statement takes, under G.C.H.,

the following form.

For any compact space X with S( X) = u+ , we have that

(a) if a is a cardinal and cf(a) does not have the form ß + with cf(/3) = u, then a

is caliber for the space X.

(b) If e = ß+ and cf(j8) = a then (a, ß) is caliber for X.

A related example shows that the result of (b) is in a sense the best possible.

Introduction. Let X be a topological space and a, ß cardinals. We say that the pair

(a, ß) is caliber for X (or X has (a, ß) caliber) if every family {Ff: £ < a) of

nonempty open subsets of X has a subfamily {V(: £ E B) such that D ies Ff ^ 0

and | B \ = ß.

Sanin in [6] has proved that if a is a regular uncountable cardinal then (a, a) (or

simply a) is caliber for a product of topological spaces iff a is caliber for each factor.

Shelah in [7] has extended this result to singular cardinals a in a smaller class,

namely the class of powers of topological spaces.

An example of Gerlits shows that Shelah's theorem fails for arbitrary products.

In this paper we study calibers of compact spaces. The main results are the

following.

Theorem 2.5. Let a be a cardinal and X a compact Hausdorff space with

S(X) = k. If a and cf(a) are strongly K-inaccessible and cf(a) > k, then a is a caliber

for X.

Theorem 3.9. Let a, k be cardinals with k < a = cf(a) and let X be a compact

Hausdorff space with S(X) — k; then (a,ia) is caliber for X.

The proofs of these theorems make use of infinitary combinatorial methods. The

main combinatorial tool is Lemma 1.1 which is a consequence of ramification

systems.

A consequence of Theorem 2.5 is an extension, under the G.C.H., of Shelah's

result in the products of arbitrary compact Hausdorff spaces.

Received by the editors August 25, 1979 and, in revised form, October 18, 1980.

1980 Mathematics Subject Classification. Primary 54A25, 05A17, 06E10.' The results of this paper were announced in Notices Amer. Math. Soc. 25 ( 1978), A-234.

2 All given terms and notation are explained in §§0 and 3.

149

© 1982 American Mathematical Society

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150 S. ARGYROS AND A. TSARPALIAS

Another application is the negative, in general, answer in the problem of the

determination of possible calibers for the class of compact spaces. More precisely

Comfort posed in [3] the problem of the determination of possible calibers on

arbitrary topological spaces. Shelah in [7] and S. Broverman, J. Ginsburg, K. Kunen

and F. D. Tall in [2] gave a positive answer to Comfort's problem. If we restrict this

problem in the class of compact Hausdorff spaces then from our results follows that

the answer is negative.

In the fourth paragraph of the paper, among others, we deal with some variations

of the above problem that possibly have a positive answer. In the same section we

prove that a class of topological spaces fails to have (a, a) caliber for a — ß+ and

cf(yß) = co. Hence we show that the estimation of Theorem 2.5 is in a sense the best

possible.

The last section mainly contains a coherent picture of the results from a set-

theoretical point of view.

0. The ordinals are defined in such a way that an ordinal is the set of smaller

ordinals. A cardinal is an ordinal not in one-to-one correspondence with any smaller

ordinal.

We call the cofinality of a cardinal a and we denote by cf(a) the least cardinal ß

such that there exists a family {a(: £ < /?} of cardinals less than a such that

a = 2(<ßa(.

A cardinal a is regular if a = cf(a) and singular if cf(a) < a. The first cardinal

strictly greater than ß is denoted by ß+ . A cardinal a is a successor cardinal if it is

of the form a = ß+ for some ß. Every successor cardinal is regular. The cardinality

of the natural numbers is denoted by to. The cardinality of a set A is denoted by

\A\. We denote by <J(a) (resp. ^„(a)) the set of "all subsets of a (resp. the set of all

subsets of a of cardinality less than k) and by <3>lc*(a) the set %(a) \ { 0 }.

The cardinality of 9(a) is denoted by 2"; more generally the cardinality of the set

of all functions from (a set of cardinality) a to (a set of cardinality) a is denoted byßa.

We set <r = 2{ax: X < k}. It is known that (ai)~ = a- for regular cardinals k.

The continuum hypothesis is the statement co+ = 2". The generalized continuum

hypothesis is the statement a+ — 2a for all infinite cardinals a.

A cardinal a is strong limit if for every ß < a: 2ß < a.

If a and k are cardinals then a is called strongly K-inaccessible if ßx < a for any

ß < a and À < k. If in addition a > k then we write a » k (or k « a). Thus e.g. if

a = (2k)+ then a » k+ ; and every infinite cardinal is strongly co-inaccessible.

The weight of a topological space X, denoted by w( X), is the least cardinal a such

that there is a base of the topology of X with cardinality a and the density character

of X, denoted by d(X), is the least cardinal ß with the property that there is a dense

subset of X with cardinality ß. The Souslin number of a topological space X,

denoted by S(X), is the least cardinal k such that there is no family of cardinality k

of pairwise disjoint nonempty open subsets of X.

The theorem of Erdös and Tarski states that the Souslin number of an infinite

Hausdorff topological space is a regular uncountable cardinal. A space X satisfies

the countable chain condition (c.c.c.) if S( X) < co+ .License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use

CALIBERS OF COMPACT SPACES 151

Let Ibe a topological space and a, ß cardinals; we say that X has precaliber

(a, ß) (caliber (a, ß)) if, for every family (F¿: £ < a) of nonempty open subsets of

X, there is B C a with \B\= ß such that the family {U(: £ E B} has the finite

intersection property (has nonempty intersection resp.). It is clear that a compact

Hausdorff space has precaliber (a, ß) if and only if it has caliber (a, ß). We say that

a space X has precaliber a (caliber a resp.) if X has precaliber (a, a) (caliber (a, a)

resp.).

1.

1.1. Lemma. Let a, p be infinite regular cardinals and k be a cardinal with

k « p < a. We also assume that for every A E a with \ A \ = a there is a partition 6SA of

A, such that \ 9A | < p. Then there is a family [An: r\ < k} of subsets of a such that

\An\= a for all r¡<K, Av+xE%ri fort]<K,

Ar,'cAv forr¡<T]'<K.

Proof. For every A C a with | A |< a, we denote by 9A the {.4} and so we extend

the notation 9A to every set A subset of a.

Inductively, for every tj < k, we define a set ct^ E 9(a) as follows.

We set 6E0 = {a}, if tj < k and <$■ has been defined, then we set

&„+x=V{%:AEé1l}.

If 7) < k is a limit ordinal and for every £ < tj &ç has been defined put

#„= ( H A(: At Effort <ji

We claim that for every tj < k, | 6? \ < p.

In fact, indeed if 7j < k and if we assume that | (2 | < p then one easily verifies that

also | i£ +I |< p.

Now, let us assume that 77 is a limit ordinal and for every £ < 77 we have that

I 6?£ I < p. Then we set ß = sup{| 6B{ | : £ < 77} and we notice that ß < 0 since p is a

regular cardinal.

We also define a map

v «, - n «c£<7J

with the rule

^(n^) = {^:£<r,}.

Obviously <i> is one-to-one and since p is strongly K-inaccessible we have that

w 11^=11 |ffi{|<i8W<p.£<7) £<i|

This completes the proof of the claim.

Setting now S, = Ur)<K6B^ we have that |6B|<p. Consequently the set &' =

{A £ &: IA I < a} also has cardinality less than p and since a is a regular cardinal it

follows that I U &' I < a. We choose an x E a \ U &'. For every r\ < k there is aLicense or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use


unique An(x) element of &, such that x E A^(x). It is easily seen that the family

{An(x): 7) < k} has the desired properties.

1.1 .a. Lemma. The conclusion of Lemma 1.1 remains true in the case a = X — k and

k is a weakly compact cardinal.

Indeed, in the construction of the sets {&a: a < k} we only need that for À, p

cardinals less than k it holds X^ < k while the existence of family {Aa: a < k) that

satisfies the conclusion of the lemma follows from properties of weakly compact

cardinals.

1.2. Lemma. Let a be an (infinite) singular cardinal, ß < a and k « a. If k = co or

k < cf(a) then there is a regular cardinal y such that ß < y < a and k « y.

Proof. If 8 < a and X < k then 8* < a; it follows that

8-= 2 Sx<a.

Now set y = ((ß-f)+ . It follows from the fact that a is not a successor cardinal

that y < a. Since y is regular, to complete the proof it is enough to show that k « y.

For 8 < y and À < k we have

s^((ß«-y-)x^((ß*-f-y- = (ß*-y-<y

(from the general set-theoretic relation ((ßs)~)s = (ß')~), as required.

1.3. Corollary. Let a be an (infinite) singular cardinal and k « a. If k = co or

k < cf(a) then there is a set {aa: a < cf(a)} such that

(i) aa is a regular cardinal for a < cf(a),

(ii) k « oía for a < cf(a),

(iii) cf(a) < a0, < aa < a for a' < a < cf(a), and

(iv)a = 20<cf(a)a„.

Proof. Since a is singular there is a family {y0: a < cf(a)} of cardinals with

properties (analogous to) (iii) and (iv). We set ß0 = y0 and recursively, if a < cf(a)

and aa, has been defined for all a' < a, we set

ßo = max{(sup{«0,: <F < o})+y„}.

By Lemma 1.2 there is aa satisfying (i) and (ii) and the condition ßa< aa < a. It is

clear that the family {aa: a < cf(a)} is as required.

2.

2.1. Lemma. Let X be a topological space with S(X) = k > co, let a be a regular

cardinal with k « a, and let {Uf £ < a} be a family of nonempty open subsets of X.

Then there is £ < a such that if ß < a and if {B^: Ç < ß] is a family of nonempty open

subsets tVj, then there is A C a, with \A\= a, such that the family {t/£ Fl Ff: £ E A)

has the finite intersection property for all f < ß.

Proof. Suppose no such £ < a exists. For A C a with | A \ = a, we choose £,, E A.

Then, there are a cardinal ßA < a and a family {B*:C<ßA} of nonempty openLicense or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use


subsets of Ut such that

If B C A, and {í/{ n F¿: £ E B) has the finite intersection property for all f < ßA,

then | F | < a.

For /I E a with | A | = a we set

•3^ = { {# E ^ : (7^ n F^ : £ E B) has the finite intersection

property for all £ < ßA}.

The family ^ partially ordered by set inclusion is inductive, hence there is a

maximal element BA E 9oA. Also it is clear that BA =£ 0 and that \BA\< a.

We define

*A:A\BA->9:{BA)XßA

as follows. Let tj E A \BA. The maximality of BA implies that there is a nonempty

finite subset Fv of BA and ^ < ßA such that

lr\u()nB?nun=0; (*)VíeF„ '

we set ^(77) = (F„, ?,,) for 77 E A \BA. We set

^ = {BA} u {«'({(F.îW^F.ne^jx/î,}

and we note that

9A is a partition of A, with | 6}A \ < a.

Since a is regular and co *£ k « a, it follows from Lemma 1.1 that there is a family

{A,: i < k] such that | A, \ = a for i < k, Ar C Ai for i < i' < k, and A¡+, £ &A for

i < K.

Thus, since | ̂ 4, | = a for all /' < k, we have

A,+ ]=4>A]({(F„J,)})

for some (F, /,) £ 9*(BA) X /L, , 1 < k. We set

Fj= ( H i/£) fl Bh for i <k,^ fei; '

and we claim that the family {V¡: i < k) consists of pairwise nonempty disjoint

subsets of X. In fact observing that for/ <k^C U( and that

è,J^A,+ x=<j>A]({Fi,Ji})

we have that

Vf) U{ = 0, and from this V,. n F- = 0 .

So the claim is true and this contradicts the assumption 5( X) = k. The proof is now

complete.

2.2. Remark. The conclusion of Lemma 2.1 remains true assuming that a — k and

a is a weakly compact cardinal. In fact applying the same methods as above and

using Lemma 1.1. a instead of Lemma 1.1 we get the desired result.License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use


2.3. Corollary. Let X be a topological space with S(X) = k > co, and let a be a

cardinal such that either a is regular with a » k or else a is weakly compact and a > k.

Then a is a precaliber for the space X.

Proof. Let {t/£: £ < a} be a family of nonempty open subsets of X. Choose £ < a

as in Lemma 2.1 or Remark 2.2. For ß = 1, F0 — i/£ we find an^Ca with | A | = a

such that the family

{(7£ n Lh: £ E A] has the finite intersection property. Thus X has

precaliber a.

2.4. Lemma. Let X be a space with S(X) = k and let a be a singular cardinal such

that X has precaliber cf(a), and

there is a family {aa: a < cf(a)} of cardinals such that aa is regular,

k « a0 for a < cf(a), a„<aTfora < t < cf(a), and 2a<:cf(a) a„ = a.

Then X has precaliber a.

Proof. Let {t/£: £ < a} be a family of nonempty open subsets of X. Since k « aa

and a0 is regular, it follows from Lemma 2.1 that there is Ja < aa such that

(*) if ß < aa and if {Fy 17 < ß} is a family of nonempty open subsets of Uj, then

there is A E aa with | A | = aa such that, for each 77 < ß, the family {t/£ fl F^: £ E ^4}

has the finite intersection property.

It follows from (*) that

(**) If 9> is a family of open subsets of ¿7, having the finite intersection property,

and if j °i) I < aa, then there is A E aa with | /I | = a„ such that the family % U {<7£: £

E /!} has the finite intersection property.

In particular,

(***) If Ja E B E aa, I B |< a„, and (i/£: £ E B] has the finite intersection prop-

erty, then there is A C aa with \Aa |= aa such that the family {l7£: £ E F U A] has

the finite intersection property.

Since cf(a) is a precaliber for X, there is S E cf(a) with | S | = cf(a) such that the

family {t/y : a E S} has the finite intersection property. Let Z = {Ja: a E S). Let B

be maximal such that Z C B C a and {£/£: £ E B) has the finite intersection

property. We claim that | B | = a. Assume the contrary; | B \ < aa for some a E S. By

(***) there is A C a0 with |/l|=a0 such that {t/£: £ E B U A) has the finite

intersection property, contradicting the maximality of B.

2.5. Theorem. Let co < k « a cz/ici k « cf(a) a«c/ /ei X be a topological space with

S( X) = k. F/je« a is a precaliber for X.

In particular, if X is a compact Hausdorff space then a is a caliber for X.

Proof. Since cf(a) is regular and k « cf(a), it follows from Corollary 2.3 that X

has precaliber cf(a). Since k « a and k < cf(a), it follows from Corollary 1.3 that

there is a family {aa: a < cf(a)} of cardinals satisfying the conditions of Lemma 2.4.

Hence X has precaliber a.License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use


2.6. Theorem. Let X be a topological space with S(X) — k > co and let a be a

cardinal such that ß- < a for every cardinal ß < a. Then a is a precaliber for X if and

only ifcf(a) is a precaliber for X.

In particular, if X is a compact Hausdorff space then a is a caliber for X if and only

if cf(a) is a caliber for X.

Proof. If a is a precaliber for X then obviously cf(a) also is a precaliber for X.

Conversely if cf(a) is a precaliber for the space X and if cf(a) < a then since

(ß~)- < a it follows that a = 20<ct(a) aa where a„ is regular aa » k. So the family

{aa: a < cf(a)} satisfies the conditions of Lemma 2.4. Hence X has precaliber a.

2.7. Corollary. Assume the Generalized Continuum Hypothesis (G.C.H.). Let X

be a topological space. Then a cardinal a is a precaliber for X if and only if cf(a) is a

precaliber for X.

In particular, if X is a compact Hausdorff space, then a is a caliber for X if and only

if cf (a) is a caliber for X.

Proof. If a — cf(a) then the result is obvious. Assuming cf(a) < a we observe

that k = S(X) =£ cf(a) and from G.C.H. for cardinals ß with cf(a) < ß < a we have

ßcf(a) _ ß+ < a an(j jience ßi < a -pije resuit follows now from Theorem 2.6.

2.8. Corollary. Assume the G.C.H. Let {Xf. i El} be a family of nonempty

spaces. A cardinal a with cf(a) > co is precaliber for X = H./e/ X¡ if and only if a is a

precaliber for X¡for all i E I.

In particular, if ' X¡ is a compact Hausdorff space, then a cardinal a with cf(a) > co is

caliber for X if and only if a is a caliber for X¡ for all i E I.

Proof. Since a is a precaliber for each X¡ it follows that cf(a) is a precaliber for

each of them, and from [6], cf(a) is a precaliber for the space X. Using Corollary 2.7

we have that a is a precaliber for X. Conversely it is clear that if a is a precaliber for

X then a is a precaliber for X¡ for all /' E F

Remark. In Corollaries 2.7 and 2.8 the G.C.H. can be replaced by the assumption

that the Singular Cardinals Hypothesis (S.C.H. in short) holds and 2cf(a) < a. The

above condition is weaker than the corresponding conditions of Corollaries 2.7 and

2.8 when we assume that cf(a) < a.

2.9. Remark. Corollaries 2.7 and 2.8 are not true for calibers of arbitrary

topological spaces in case a is a singular cardinal as follows from the example given

in [4].

2.10. Corollary. Assume the G.C.H. Let a be infinite cardinal and X a topological

space with S(X) — k > co.

(a) // cf(a) > k, and if either cf(a) is a limit cardinal or else cf(a) = ß+ with

cf(ß) > k, then a is a precaliber for X.

(b) Ifcî(a) < k then a is not precaliber for X. Analogous statements for calibers hold

in case X is a compact Hausdorff space.

Remark. The remaining cases in the above theorem, namely either cf(a) = k or

else cf(a) = ß+ with cf(/F) < k, will be investigated in the next sections.

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2.11. Corollary. Assume the Generalized Continuum Hypothesis. Let X be a

topological space with S(X) = ux.An infinite cardinal a is a precaliber for X, unless

(a) cf(a) = co

in which case a is not a precaliber for X; or

(b) cf(a) = ß+ , with cf()S) = co

in which case a may or may not be a precaliber for X.

The analogous statement for calibers holds in case X is a compact Hausdorff space.

Proof. This follows from the case k = co+ of Corollary 2.10.

3.

3.1. Definition. Let a, k be infinite cardinal numbers. We set

{a = min{/8: ßK > a), fa = min Na: X < k .

It is clear that if X < k then 2 < fa < yä~ < a; also

"{a = min{/J: there is X < k with ßx > a).

3.2. Remark. If fä < ß ^ a then fa = 1ß. (Indeed it is clear that K/ß^fä; if

Ky[ß < fä then fä *£ Cfß)K < a, and hence

(To")" < (7ß)K" = (7ß)K < «,

which is impossible.)

3.3. Proposition. Let a, k be infinite cardinals and suppose that -fa < a. Then either

fa = 2 or else, setting ß = 7ä, we have cf(/3) < k < ß.

Proof. Suppose vS" > 2, and set ß = ^5". Then ß > k. We prove that cf(jS) =s k.

Indeed, suppose that cf(/F) > k. Let y be a regular cardinal such that \fa < y ^ a; by

the above remark, we have 1/y = ß. Since ßK s* y, it follows that there is a one-to-one

function/: y — ß. Define <t>: y -» ß by <í>(£) = suprange(/(£)(K)). Since cf(^) > k,

<j> is well defined; since y is regular and ß < y, it follows that there are tj0 < ß and

A E y, with \ A |= y, such that <f>(£) < Vo lor all £ E A; hence

f\A: A ^K7)0. Hence y =|/4 |<|t}0|k, and ß >| t}0 \> Kjy , a contradiction.

It now follows that cf(j8) < k < ß.

3.4. Corollary. Let a, k be infinite cardinals and suppose that fa < a. Then either

fä = 2 or else, setting ß = %ä~ we have cf(/3) < k.

3.5. Remark. It is clear that if a, k are cardinals such that fä > co, then va" is a

strongly k inaccessible cardinal. Furthermore, if we assume that k is a regular

cardinal and co *s T'a < « then fa » k.License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use


3.6. Lemma. Let X be a space with S(X) = k > co, let a be a regular cardinal such

that 2 < fä < a, and let {Uf £ £ a} be a family of nonempty open subsets of X. Then

there is A C a with \ A \ = a such that: if A C A, | A | < k£ and [Uf £ E A} has the

finite intersection property, then

|{ij£./4:{í/£:£eAU (tj}} has the finite intersection property] \ = a.

Proof. Suppose that there is no such set A. For A C a with \A | = a, there are

BA E A with | BA |< a and AA C A with | A^ |< k- such that {t/£: £ E A,,} has the

finite intersection property, while for all 77 E A \BA the family {Uf £ E AA U {rj}}

does not have the finite intersection property. We define

*A:AsBA^9:(AA)

as follows. Let 17 E A \ BA; there is a nonempty finite subset Fn of AA such that

Pi t/{ n u„ = 0 ;

we set ^(tj) = Fv for 77 E A \ BA. We set

% = {BA} U {4>A\{F}): F E$:(AA)}

and we note that

<$A is a partition of A, with \9A | =s k^ .

We now apply Lemma 1.1, with p = (k-)+ ; since k is regular it follows that k « p

and since a is regular and fä > 2, it follows that p < a; thus, there is a family

{Af: i < k) such that | A,■ | = a for / < k, Ar E y4; for i < i' < k, and Al+, E ÍP^ for

/' < k. Thus, since | A¡ \ = a for all /' < k, we have

¿m = «!({*;})

for some F, E i?w*( A^ ), /' < k. We set

V,= Di/j for i < k,

and it is easy to verify that the family {V¡: i < k} consists of pairwise disjoint

nonempty open subsets of X. This is a contradiction to the fact that S(X) = k.

3.7. Lemma Let a, k be infinite cardinals with k regular, cf(a) < k, and a strongly

K-inaccessible. Then there is a family {aa: a < cf(a)} of cardinals such that aa is

regular, aa < a, aa is strongly K-inaccessible for a < cf(a), and a = 20<cf(a) aa.

Proof. Let {/?„: a < cf(a)} be a family of regular cardinals, ßa < a and a =

2"a<cf(a) Pa-

For every a < cf(«) we claim that ß% < a.

In fact assuming that this is false, then there is a0 < cf(a) such that ß- > a. But

then we will have ß^ > a. (Really, if ßj = a since k is a regular cardinal we have

that

a — ß^g- (ß;0)~ - a-ac!(tt) > a, a contradiction.)

That contradicts the fact a is strongly K-inaccessible and this proves the claim.License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use


Now setting aa = (ß*)+ , we can easily verify that the family [aa: a < cf(a)}

satisfies the conclusion of the lemma.

Remark. Combining Corollary 1.3 and Lemma 3.7 we observe that if a is strongly

K-inaccessible cardinal and k is regular uncountable cardinal and cf(a) ¥= k, then

always there is a family {aa: a < cf(a)} of regular strongly K-inaccessible cardinals

with 2 aa = a. In the remaining case cf(a) = k the above family exists if k is a

successor cardinal while in case k is a limit cardinal it does not exist.

3.8. Definition. Let A" be a topological space, and let % = {(7£: £ E a} be a

family of nonempty open subsets of X. An index £ < a is called a representative of

% if the following condition is satisfied:

If ß < a and if {Br: f < ß) is a family of nonempty open subsets of U¡, then there

is A C a, with \A \ — a, such that the family {t/£ n B¡: £ £ A] has the finite

intersection property for all f < ß.

Thus, Lemma 2.1 says that, if S(X) = k « a, and a is regular, then every family

% = {t/{: £ £ a} of nonempty open subsets of X has a representative.

3.9. Theorem. Let X be a space S( X) = k > co, and let a be a regular cardinal with

k < a. Then X has precaliber (a, co • fä).

Proof. If fä = 2, then according to Rosenthal's result [5], A" has precaliber (a, co).

If fä — a, then a is regular and a » k, and hence from Theorem 2.5, X has

precaliber (a, a).

So we assume from now on that 2 < fä = ß < a. Let {(7£: £ < a] be a family of

nonempty open subsets of X. We note that ß » k and cf(/T) < k, by Corollary 3.4

and Remark 3.5, and hence from Lemma 3.7, there is a family [ßa: a < cf(/?)} of

cardinals such that

A, is regular, ßa<ß, k « ßjor a < cf(ß),

and

0= 2 A-o<cf(j8)

Claim. There is a family {/10: a < cf(/T)} of subsets of a such that | Aa \ = ßa and

the family {%£: £ £ Aa) has a representative £a such that the family {t/£ : a < cf(/8)}

has the finite intersection property.

Indeed, let A be a subset of a with | .4 | = a and the property of Lemma 3.6. We

proceed inductively. Let a < cf(/T). We assume that AT, £T have been defined for

T < a such that AT E A, \ AT \ = ßT, {%£: £ E AT) has the finite intersection property.

Let A = {£T: t < a}; thus | A |< cf(/?) < ks; hence, from the characteristic prop-

erty of A, as in Lemma 3.6, it follows that the set

B = (t)E/1:{í/£:£eAU {77}} has the finite intersection property}

has cardinality a.

We choose Aa E B with \A„ |= ßa and a £„ in Aa such that {%£: £ E Aa] has

representative £a. So the inductive definition is complete.License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use


Now setting Ax = U „<cf(a) Aa we claim that there is A2 E Ax with \A2\= ß such

that the family (¿7£: £ E A2} has the finite intersection property. Indeed, let C be a

maximal subset of Ax such that {£„: a < cf(ß)} C C and {i/£: £ £ C} has the finite

intersection property. It is enough to show that \C\> ß. Suppose on the contrary

that | C\< ß. Then | C\< ßa for some a < cf(/?). Now the fact that £„ E C is a

representative of (<7£: £ E Aa) allows us to find 77 E Aa\C such that {t/£: £ E C U

{77}} has the finite intersection property, contradicting the maximality of C.

Incidentally we give a simple proof of Rosenthal's result which states that

3.10. Proposition. // a is a regular cardinal and a > S(X), then X has (a, u)

precaliber.

Proof. Let {C/£: £ E a) be a family of nonempty subsets of X. Inductively we

choose sequences

{í/£|,(7£2,...,í/£n,...,«<co}, {Ax,A2,...,A„,...,n<(o}

such that

(i) a D Ax D A2D ■ ■ ■ D AnD ■ ■ ■ n < co and | An | = a,

(ii) PI "i=x U(i n(/^0 for all s E A„.

This choice is possible from the regularity of a and the fact that a > k = S( X).

Now the family {f/£ : n < co} obviously has the finite intersection property.

4.

4.1. Theorem. Let ß be a strong limit cardinal with cf(/8) = co, and assume that

2ß — ß+ . Then there is an extremally disconnected compact Hausdorff space Í2 such

that ß has a strictly positive measure (and hence, satisfies the c.c.c.) and Ü does not

have caliber ß+ .

Proof. Let X be the product measure that is inducted on the space (0,1}^ by the

coordinate measures (A£: £ < /?}, where X£({0}) = A£({1}) = {- for all £ < ß. Let

also S be the a-algebra of X-measurable subsets of {0,1 }ß.

Claim 1. If A is a subset of (0,1}^ with \A \< ß then A E S and X(A) = 0.

Indeed, since cf(ß) = u it follows that A = U n<uAn such that \An |< ß for all

« < co, and hence it is enough to prove the claim for sets A with | A | < ß.

Let A = {xa: a < y}, y < ß, be a well ordering of A and for f < ß we set At E A

such that Aç = {xa: xa(Ç) = 0}. Since ß is strong limit it follows that 2,1'l< ß and

thus there is DA C ß such that \DA\>u and if f„ f2 E DA, ASi = Afj. Let B =

{xx, x2} X {0,1}^°" where xx(Ç) = 0, x2(?) = 1 for all f E DA. We easily verify

that A C B and from well-known facts on properties of product measure it follows

thatA(F) = 0.

We next claim that there is a family {K^: 17 < ß+ } of subsets of {0,1 }ß such that

Kv is compact for 77 < ß+ , X(Kv) > 0 for 77 < ß+ , and if A C ß+ , \ A \ = ß+ then

the family {K^: 77 E A} does not have the finite intersection property.

Indeed, let {xc: £ < ß+ } be a one-to-one enumeration of {0,1}^, using the

hypothesis 2ß = ß+ . We note that the set {x£: £ < 77} has cardinality | tj | < ß for all



77 < ß+ and hence from the statement proved above

a({jc£:£<77}) = 0 for all 77 </3+.

By the regularity of (the Haar measure) X for t) < ß+ there is a compact subset

Kn E {jc£: £ > 77} with X(Kv) > 0. Let A C ß+ with | A | = ß+ , then it is easy to

verify that iliE/,= 0.

Now for 77 < ß+ let [Kn] denote the ^equivalence class of Kv and Í7, denote the

unique open-and-closed subset of ß that corresponds to [K ] via Stone's duality

theorem.

Let A E ß+ with I j4 | = ß+ . We claim that [U : 77 E A} does not have the finite

intersection property. This is equivalent to the claim that {[A' ]: 77 E A} does not

have the finite intersection property. This follows from the fact that D A K^ = 0.

The proof is complete.

4.2. Corollary. Assume the Generalised Continuum Hypothesis, and let a be a

cardinal such that cf(a) — ß+ and cf(ß) = co.

Then there is an extremally disconnected compact Hausdorff space ß with the c.c.c.

such that

ß does not have caliber a, but has caliber (a, ß).

4.3. Remark, (a) Corollary 4.2 shows that Theorem 3.9 is the best possible (at

least assuming the G.C.H.).

(b) The above example extends a well-known example of Erdös on nonexistence of

co+ caliber for all c.c.c. compact spaces. Namely Erdös has proved that under the

G.C.H. the Stone space of the Boolean algebra of measurable subsets of [0,1] with

Lebesgue measure modulo null sets fails to have co+ -caliber.

(c) It follows from Theorem 4.1 that (at least assuming the G.C.H.) there are

compact spaces ß satisfying the c.c.c. and that fail to have caliber for arbitrary large

regular cardinals.

(d) In [1] an extension of Theorem 4.1 that has the following form is given:

Theorem. Assume the G.C.H. Then for every a, k cardinals with cî(a) < k there is

an extremally disconnected space ßa such that S(£la) < k and ßa does not have a +

caliber.

4.4. Remark. Let A be a compact Hausdorff space, p a regular Borel measure on

X, and a a cardinal such that co+ < a and co+ « cf(a), and (S£: £ < a} a family of

p-measurable subsets of X with p(S£) > 0 for £ < a. Then there is A E a with

I A |= a and D ieASi¥= 0.(Indeed, let ß be the Stone space of the measure algebra L — <3lt/9t of the

measure p. Then ß has a strictly positive measure, hence satisfies the c.c.c, and thus

ß has caliber a (Theorem 2.5). From the regularity of the measure p there is a

compact set #£ such that ÄT£ E S£ and p(K()> 0, for £ < a. The p-equivalence class

[FT£] corresponds by Stone's duality to a (unique) nonempty open-and-closed subset

of ß. Hence there is^Ca with \A \= a such that {[K(]: £ £ A} has the finite



intersection property and since each Kf is a compact subset of X it follows that

H teA 5£ D n £6/j Íj ^ 0 as required.)

4.5. Remark. Theorem 2.5 implies a negative, in general, answer to the Comfort

problem on the determination of possible calibers for compact spaces. Indeed, let

H = {ay: y E T} be a family of regular cardinals, co E H and if we want to find a

compact space XH that does not have caliber exactly on the cardinals a such that

cf(a) E H, then the family H need satisfy the following condition. If k is the least

regular cardinal that does not belong to H, then obviously S( X) < k and therefore

from Theorem 2.5 it follows that every regular a such that a » k also does not

belong to H.

However we do not know if for a family H that satisfies the above condition

always there is the space XH. In particular we do not know even under the G.C.H. if

for a given family H of regular cardinals such that co+ £/i and for a E H, a has the

form a = ß+ , cf(ß) = co, there is the compact space XH.

5. This section is due to the referee and contains a coherent picture of what has

been proved and what cases are left open.

5.1. Definition. For infinite cardinals a, ß, k let the symbol a => (k, ß) denote the

statement: every space X with S( X) =£ k has precaliber (a, ß).

Let f (k, ß) - min{a: a => (k, ß)}.

5.2. About function f. Some properties of the function f are the following:

(a) f (k, ß) is monotonically increasing in both arguments.

(b) f (co, ß) — ß for every cardinal ß.

(c) f(co, ß) = ß for every ß (Proposition 3.10).

(d) If k is singular then f (k, ß) = sup{f(X, ß): X < k} (Erdös-Tarski theorem).

From now on, we assume for simplicity the G.C.H.; by Theorem 3.9 if y =

max{K, ß) then y < f(k, ß) =s y+ . We deal with the problem of deciding between y

and y+ .

Case 1. ß < k. Then f (k, ß) = k if ß = co or if k is singular. If co < ß < k = cf(K),

we do not know anything about the values of í(k, ß).

Case 2. /} = k. f(K, k) = k if k is a weakly compact cardinal (for example if k = co)

while f(K, K) = K+ifKisa singular cardinal or a successor cardinal or if there is a k

Souslin tree. Thus (by Jensen's result) assuming V = L, we have f (k, k) = k if and

only if k is weakly compact. We do not know if this is true assuming only G.C.H.

Case 3. ß > k. If k = co then f(K, ß) = ß. If cf(ß) < k then f(K, ß) = ß+ . If

cf(ß) = k then f(K, ß) = ß if and only if f(K, k) = k (Case 2). If cf(/?) > k and if

cf(/T) is a limit cardinal or else cf(/6) = y+ with cf(y) > k then f(K, ß) = ß. If

cf(/?) = y+ and co =£ cf(y) < k then by Theorem 4.1 and [1] (see Remark 4.3(d)) it

follows that f(K,/?) = /?+.

Finally, the following statements are easily seen to be equivalent for all cardinals

a, ß and k.

(1) Every extremally disconnected compact Hausdorff space X with S(X) =s k has

caliber (a, ß).

(2) Every space X with S( X) *£ k has precaliber (a, ß).



(3) Given any family {<7£: £ E a) of nonempty sets, either there is B E a such that

| B | = ß and {t/£: £ E B} has the finite intersection property, or else there is a family

{Vv: 7j < k} of pairwise disjoint nonempty sets such that each V is the intersection

of finitely many i/£'s.

(4) For every family £ E 9(a) with 2 < | X | < co for each X E £, either there is an

independent set B C a with \B\= ß, or else there is a family {^4£: £ < k} of finite

independent subsets of a such that A¡_ U A^ is dependent for £ < 77 < k. (A set

A C a is "independent" if £ n 9(A) = 0 ; "dependent" if til 9(A) ¥= 0.)

Note that if we weaken (4) by requiring |A"|=2 for X E £, it becomes a

weakened version of the ordinary partition relation a -> (k, ß)2 of Erdös and Rado.

Acknowledgment. We would like to express our thanks to Professor S. Negre-

pontis for his great help during the preparation of the paper. We are also grateful to

the referee for his valuable suggestions on the material in it. His numerous

comments helped us to improve parts of this paper; among others, following his

suggestions, we simplified the proofs of Lemmas 1.1 and 2.4.

Added in proof. Corresponding results for independent families of sets and their

applications to Topology and Functional Analysis, proved by the authors, are

contained in the paper, "Embeddings of l\T) into subspaces of C(ß)", to appear in

the Mathematical Proceedings of the Cambridge Philosophical Society. Similar

results have been established by J. Shelah in Remarks on Boolean algebras, Algebra

Universalis 11 (1980), 77-89.

References

1. S. Argyros, Boolean algebras without free families, Algebra Universalis (to appear).

2. S. Broverman, J. Ginsburg, K. Kunen and F. D. Tall, Topologies determined by a-ideals on to,, Canad.

J. Math. 6 (1978), 1306-1312.

3. W. W. Comfort,/! survey of cardinal invariants. General Topology Appl. 1 (1971), 163-199.

4. J. Gerlits, manuscript notes, 1978.

5. H. P. Rosenthal, O77 injective Banach spaces and the spaces L°°(/i) for finite measures ¡x. Acta Math.

124(1970), 205-248.6. N. A. Sanin, On the product of topological spaces, Trudy Mat. Inst. Steklov. 24 ( 1948), 1-112.

7. S. Shelah, Remarks on cardinal invariants in topology. General topology Appl. 7 (1977), 251-259.

Chair I of Mathematical Analysis, University of Athens, Athens, Greece (621)


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