Tomek Bartoszynski, Saharon Shelah and Boaz Tsaban- Additivity Properties of Topological Diagonalizations

8/3/2019 Tomek Bartoszynski, Saharon Shelah and Boaz Tsaban- Additivity Properties of Topological Diagonalizations

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ADDITIVITY PROPERTIES OF TOPOLOGICAL

DIAGONALIZATIONS

TOMEK BARTOSZYNSKI, SAHARON SHELAH, AND BOAZ TSABAN

Abstract. In a work of Just, Miller, Scheepers and Szeptycki itwas asked whether certain diagonalization properties for sequencesof open covers are provably closed under taking finite or countableunions. In a recent work, Scheepers proved that one of the prop-erties in question is closed under taking countable unions. Aftersurveying the known results, we show that none of the remain-

ing classes is provably closed under taking finite unions, and thussettle the problem. We also show that one of these properties isconsistently (but not provably) closed under taking unions of sizeless than the continuum, by relating a combinatorial version of thisproblem to the Near Coherence of Filters (NCF) axiom, which as-serts that the Rudin-Keisler ordering is downward directed.

1. Introduction

1.1. Selection principles. Let U and V be collections of covers of a

space X. The following selection hypotheses have a long history forthe case when the collections U and V are topologically significant.

S1(U,V): For each sequence {Un}nN of members ofU, there is a sequence{Vn}nN such that for each n Vn Un, and {Vn}nN V.

Sfin(U,V): For each sequence {Un}nN of members ofU, there is a sequence{Fn}nN such that each Fn is a finite (possibly empty) subsetof Un, and

nN Fn V.

Ufin(U,V): For each sequence {Un}nN of members of U, there is a se-quence {Fn}nN such that for each n Fn is a finite (possibly

1991 Mathematics Subject Classification. Primary: 37F20; Secondary 26A03,03E75 .

Key words and phrases. Menger property, Hurewicz property, selection princi-ples, additivity numbers, Rudin-Keisler ordering, near coherence of filters.

The first author is partially supported by the NSF grant DMS 9971282 andAlexander von Humboldt Foundation. The research of the second author is partiallysupported by The Israel Science Foundation founded by the Israel Academy ofSciences and Humanities. Publication 774.

This paper constitutes a part of the third authors doctoral dissertation at Bar-Ilan University.

1


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2 TOMEK BARTOSZYNSKI, SAHARON SHELAH, AND BOAZ TSABAN

empty) subset of Un, and either for some n Fn = X, or else{Fn}nN V.

Assume that V U. Following [29], we say that X satisfiesUV

(read: U choose V) if for each cover U U there exists a subcoverV U such that V V. Observe that Sfin(U,V) implies

U

V

.

1.2. Special covers. We will concentrate on spaces for which theusual induced topology has a subbase whose elements are clopen (bothclosed and open), that is, sets which are zero-dimensional. More specif-ically, we will usually work in P = R \Q, and by set of reals we meana subset ofP (or any homeomorphic space).

Let X be a set of reals. An -cover U of X is a cover such that

each finite subset of X is contained in some member of the cover. Uis a -cover of X if it is infinite, and each element of X belongs toall but finitely many members of the cover. Let O, , and de-note the collections of countable open covers, -covers, and -coversof X, respectively, and let B, B, B be the corresponding countableBorel covers. The diagonalization properties for these types of coversinclude as particular cases the properties Ufin(, O) of Menger [19],Ufin(, ) of Hurewicz [15], S1(, ) of Gerlits and Nagy (known asthe -property) [14], S1(O, O) of Rothberger (known as the C prop-erty) [21], Sfin(, ) of Arkhangelskii [1], and S1(, ) of Sakai [22].These properties were extensively studied in the general framework in,

e.g., [23, 17, 26]. Many of these properties turn out equivalent. Thesurviving properties appear in Figure 1, where an arrow denotes im-plication. To understand the implications in this diagram, use the factthat S1(O, O) = S1(, O) [23]. Only two implications (the dotted onesin the diagram) remain unsettled.

In the diagram, each property appears together with its critical car-dinality, that is, the minimal size of a set of reals which does not satisfythat property. The constants p, b, d, s, and cov(M) are the pseudo-intersection number, the unbounding number, the dominating number,the splitting number, and the covering number of the meager (first cat-egory) ideal, respectively see, e.g., [12, 8] for the definitions of theseconstants.

1.3. Additivity properties. One of the important questions about aclass J of sets of reals is whether it is closed under taking countableor at least finite unions, that is, whether it is countably or finitelyadditive. A more informative task is to determine the exact cardinality such that a union of less than members of J belongs to J, buta union of many need not. This cardinal is called the additivity


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ADDITIVITY OF TOPOLOGICAL DIAGONALIZATIONS 3

Ufin(,)b

GG?

55

Ufin(,)d

GG?

Ufin(,O)d

Sfin(,)d

YYvvvv

S1(,)b

GG

XXuuu

uuuuuuuuuuuu

S1(,)d

GG

YYvvv

v

S1(,O)d

bb~~~

~~~~~~~~~~

S1(B,B)b

``xxxx

GG S1(B,B)d

WWtttt

GG S1(B,B)d

YYxxx

x

Sfin(,)d

yy

Sfin(B,B)d

yy

UUpppppppppppp

S1(,)p

GG

yy

S1(,)cov(M)

yy

GG

pp

S1(O,O)cov(M)

yy

S1(B,B)p

yy

GG

aazzzz

S1(B,B)cov(M)

yy

XXuuuu

GG S1(B,B)cov(M)

yy

``zzzz

Figure 1. The surviving classes

number of J, and denoted add(J). Despite the extensive study of theclasses defined by the mentioned diagonalization properties, not muchwas known about their additivity properties. The purpose of this paperis to clear up most of what was not known with respect to the additivitynumbers of these properties. It turns out that most of the classes arenot provably closed under taking finite unions. For some of the classeswhich are not provably closed under taking finite unions, we considerthe consistency of their being closed under taking finite (and larger)unions. For S1(B, B) and Ufin(, ) this turns out to be related tothe well known NCF (Near Coherence of Filters) axiom, which asserts

that the Rudin-Keisler ordering of ultrafilters is downward directed.

2. Positive results

In [17] it is pointed out that the properties Ufin(, O) (Mengersproperty), Ufin(, ) (Hurewicz property), S1(O, O) (Rothbergers prop-erty), and S1(, O) are closed under taking countable unions. Theargument behind these assertions actually shows the following.


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Proposition 2.1. Each property of the form (U, O) (or (U, B)), {S1, Sfin, Ufin}, is closed under taking countable unions.

Proof. Let A1, A2, . . . be a partition of N into disjoint infinite sets.Assume that X1, X2 . . . satisfy (U, O). Assume that {Un}nN Uare covers of X =

kN Xk. For each k, use this property of Xk to

extract from the sequence {Un}nAk the appropriate cover Vk of Xk.Then

kN Vk is the desired cover of X.

The proof for (U, B) is identical.

We now give some more exact additivity results. Denote the criticalcardinality of a class J of sets of reals by non(J). The covering numberofJ, cov(J), is the minimal cardinality of a subcollection F J suchthat F = P. Then add(J) is a regular cardinal, add(J) cf(non(J)),and add(J) cov(J).

Proposition 2.2. If I and J are collections of sets of reals such that:

X I if, and only if, for each Borel function : X P[X] J.

thenadd(J) add(I).

Proof. Assume that Xi, i I, are members of I such that X =iIXi I. Then there exists a Borel function : X P such

that [X] J. But [X] =

iI[Xi].

Using results from [26] and [30], we get from Proposition 2.2 thatthe following inequalities hold.

(1) add(S1(O, O)) add(S1(B, B)) cf(cov(M)),(2) max{add(S1(, )), add(Ufin(, ))} add(S1(B, B)) b,(3) max{add(S1(, O)), add(Ufin(, O))} add(S1(B, B)) cf(d),(4) add(S1(, )) add(S1(B, B)) p;(5) max{add(S1(, )), add(Sfin(, )), add(Ufin(, ))}

add(S1(B, B)) cf(d).

LetN denote the collection of null (measure zero) sets of reals. In [3]it is proved that add(N) add(S1(O, O)). In order to get similar lowerbounds on other classes, we will use combinatorial characterizations of

these classes. The Baire space NN is assigned the product topology.Hurewicz ([16], see also Reclaw [20]) proved that a set of reals X sat-isfies Ufin(, O) if, and only if, every continuous image of X in NN isnot dominating. Likewise, he showed that X satisfies Ufin(, ) if, andonly if, every continuous image of X in NN is bounded (with respect to). It is easy to see that a union of less than b many bounded subsetsofNN is bounded, and a union of less than b many subsets ofNN whichare not dominating is not dominating.


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Corollary 2.3. The following holds:

(1) add(Ufin(, )) = add(Ufin(B, B)) = b;(2) b add(Ufin(, O)) add(Ufin(B, B)) cf(d).

Consider an unbounded subset B of NN such that |B| = b, anddefine, for each f B, Yf = {g NN : f g}. Then the sets Yfare not dominating, but

fB Yf =

NN: For each g NN there existsf B such that f g, that is, g Yf. Thus the second assertion inCorollary 2.3 cannot be strengthened in a trivial manner.

Problem 2.4. Is add(Ufin(, O)) provably equal to b?

Let h be the density number [8]. Then 1 h. Recently, Scheepers[24] proved that h add(S1(, )). Thus, h add(S1(, )) b.

3. Negative results

Showing that a certain class is not closed under taking finite unionsis apparently harder: All known results require axioms beyond ZFC.(This is often necessary see Section 4.) In [13, 29] it is shown thatassuming the Continuum Hypothesis, no class between S1(B, B) andS1(, ) is closed under taking finite unions.

The following problem is posed in [17].

Problem 3.1 ([17], Problem 5). Is any of the classes S1(, ), S1(, ),S1(, ), Sfin(, ), Sfin(, ) andUfin(, ) closed under taking count-

able or finite unions?As mentioned in Section 2, Scheepers answered the question posi-

tively for S1(, ). We will show that if cov(M) = c (in particular,assuming the Continuum Hypothesis), then the answer is no for all ofthe remaining classes. In fact, we will show that none of the classeswhich lie between S1(B, B) and Ufin(, ) is provably closed undertaking finite unions.

For clarity of exposition, we will first treat the open case, and thenexplain how to modify the constructions in order to cover the Borelcase.

3.1. The open case. For convenience, we will work in NZ (with point-wise addition), which is homeomorphic to P. The notions that we willuse are topological, thus the following constructions can be translatedto constructions in P.

A collection J of sets of reals is translation invariant if for each realx and each X J, x + X J. J is negation invariant if for eachX J, X J as well. For example, M and N are negation andtranslation invariant (and there are many more examples).


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Lemma 3.2. If J is negation and translation invariant and if X is aunion of less than cov(J) many elements of J, then for each x NZ

there exist y, z NZ \ X such that y + z = x.

Proof. (x X) X is a union of less than cov(J) many elementsof J. Thus we can choose an element y NZ \ ((x X) X) =(x NZ \ X) (NZ \ X); therefore there exists z NZ \ X such thatx z = y, that is, x = y + z.

For a finite subset F ofNN, define max(F) NN to be the functiong such that g(n) = max{f(n) : f F} for each n. A subset Y of NN,is finitely-dominating if the collection

maxfin(Y) := {max(F) : F is a finite subset of Y}

is dominating. Y is k-dominating if for each g NN there exists ak-element subset F of Y such that g max(F) [7]. Clearly each k-dominating subset of NN is also finitely dominating. Following is thekey observation for our solution of Problem 3.1.

Theorem 3.3 ([30]). For a set of reals X, the following are equivalent:

(1) X satisfies Ufin(, );(2) For each continuous function from X to NN, [X] is not

finitely-dominating.

Proposition 3.4. Assume thatcov(M) = c. Then there exist c-Luzin

subsets L0 and L1 ofN

Z satisfying S1(, ), such that the (c-Luzin)set L0 L1 is 2-dominating. In particular, L0 L1 does not satisfyUfin(, ).

Proof. This is a generalization of the constructions of [17] and [26].Assume that cov(M) = c. Let {y : < c} enumerate NZ; let {M : < c} enumerate all F meager sets in NZ (observe that this familyis cofinal in M), and let {{Un }nN : < c} enumerate all countablesequences of countable families of open sets.

Fix a countable dense subset Q NZ. We construct L0 = {x0 : 1.

Fix i < 2. For each n, define

fi(n) =

g(a(k 1)) + n a(k 1) n [a(k), a(k + 1)) for k Ai

g(a(k)) + n a(k)

n [a(k), a(k + 1))

where k Ai, k + 1 Aig(n) otherwise

It is not difficult to verify that fi NN.For each k Ai and n [a(k), a(k + 1)),

fi(n) = g(a(k 1)) + n a(k 1)

a(k) + n a(k 1) a(k) + n 2n h(n).

Therefore fi YFi,h.For each n let k be such that n [a(k), a(k+1)). Ifn is large enough,

then either k, k + 1 A1, and therefore f1(n) = g(n), or else k, k + 1 A2, and therefore f2(n) = g(n), that is, g(n) max{f1(n), f2(n)}.

This completes the proof of Theorem 4.4.

The following two lemmas will imply that if NCF holds, then Dfinis closed under taking countable, and consistently much larger, unions.For each k N, let Dk be the collection of subsets ofNN which are notk-dominating.

Lemma 4.6. Assume that |I| < b and Yi NN, i I, are such that for eachF I with |F| k,

iFYi Dk. TheniI

Yi Dk.

Proof. For each F I with |F| k, Let gF NN witness thatiF Yi Dk. The collection {gF : Finite F I} has size < b,

therefore it is bounded, say by h NN. Then h witnesses that Y =iIYi Dk: Each k-subset Z of Y is contained in

iF Yi for a suit-

able k-subset F of I. Thus max(Z) cannot dominate gF; in particularit cannot dominate h.

Let g be the groupwise density number [7, 8].


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Lemma 4.7 (Blass [7]). Assume that |I| < g and Yi NN, i I,are such that for each F I with |F| k, iF Yi D2k. Then

iIYi Dk.

Corollary 4.8. IfDfin is closed under taking finite unions, then it isclosed under taking unions of size < max{b, g}.

Proof. Assume that for each i I, Yi Dfin. Then for each k, andeach finite F I with |F| k,

iF Yi Dfin D2k Dk. Thus,

if |I| < b (respectively, |I| < g), then by Lemma 4.6 (respectively,Lemma 4.7)

iIYi Dk for all k. As Dfin =

kDk [30], we have that

iIYi Dfin.

The ultrafilter number u is the minimal size of an ultrafilter base.

It is known that u < g implies NCF; the other direction is open: Allthe known models of NCF [10, 11] satisfy u < g. In particular, it isconsistent that u < g. u < g also implies that b = u < g = d = c [8].

Corollary 4.9. Ifu < g holds, thenDfin is closed under taking unionsof size < c.

We can now summarize the results of this section.

Theorem 4.10. The following are equivalent:

(1) NCF,(2) D

finis closed under taking finite unions;

(3) Dfin is closed under taking unions of size < max{b, g}.

Corollary 4.11. (1) If NCF holds, then

max{b, g} add(Ufin(, )) add(S1(B, B)) cf(d) = d.

(2) Ifu < g, then

add(Ufin(, )) = add(S1(B, B)) = c > u.

Proof. (1) follows from the inequalities of Section 2, Corollary 4.1, andTheorem 4.10 (together with the fact that NCF implies that d is regular[5]).

(2) follows from (1), as u < g implies g = c.

Problem 4.12. Is any of the classes Sfin(, ), S1(, ), andSfin(, )consistently closed under taking finite unions?

For the Borel case there remains only one unsolved class.

Problem 4.13. Is Sfin(B, B) consistently closed under taking finiteunions?


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5. -covers

U is a -cover of X if it is a large cover of X (that is, each memberof X is contained in infinitely many members of the cover), and foreach x, y X, (at least) one of the sets {U U : x U, y U} and{U U : y U, x U} is finite. -covers are motivated by the towernumber t [28] and were incorporated into the framework of selectionprinciples in [29]. Let T and BT denote the collections of countableopen and Borel -covers of X.

By Proposition 2.1, S1(T, O) and S1(BT, B) are also closed under tak-ing countable unions. In [29] it is shown that assuming the ContinuumHypothesis, no class between S1(B, B) and

T

is closed under taking

finite unions (recall that Sfin(, T) implies T). It is also proved there

that add(S1(BT, B)) = t. Using Scheepers result mentioned in Section2, we have that this also holds in the open case.

Theorem 5.1. add(S1(T, )) = t.

Proof. Clearly S1(T, ) =T

S1(, ). In [29] it is shown that

add(T

) = t. Thus, t = min{t, h} add(T, ) non(S1(T, )) =

t.

Problem 5.2. Is any of the properties S1(T, T), Sfin(T, T), S1(, T),Sfin(, T), andUfin(, T) (or any of their Borel versions) provably (orat least consistently) closed under taking finite unions?

Problem 5.3. Is any of the classes Sfin(, T), S1(T, ), andSfin(T, )consistently closed under taking finite unions?

Remark 5.4. 1. It turns out that in [25], Scheepers used the Con-tinuum Hypothesis (a stronger assumption then our cov(M) = c) toconstruct two sets satisfying S1(, ) such that their union does notsatisfy Sfin(, ). This is extended by our Proposition 3.4, which inturn is extended by Theorem 3.5.

2. Our paper was written in 2001, more or less the time Blass paper[9] was written. This explains some overlaps between the results of that

paper and the current one. In particular, Blass proves there a slightlyweaker version of Theorem 4.10 see Corollary 5.5, Theorem 6.5, andTheorem 6.6 in [9].

References

[1] A. V. Arkhangelskii, Hurewicz spaces, analytic sets, and fan tightness of func-tion spaces, Soviet Mathematical Doklady 33 (1986), 396399.

[2] T. Bartoszynski, Combinatorial aspects of measure and category, FundamentaMathematicae 127 (1987), 209213.


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[3] T. Bartoszynski and H. Judah, On the smallest covering of the real line by mea-ger sets (II), Proceedings of the American Mathematical Society 123 (1995),

18791885.[4] T. Bartoszynski and B. Tsaban, Hereditary topological diagonalizations and

the Menger-Hurewicz Conjectures, Proceedings of the AMS, to appear.http://arxiv.org/abs/math.LO/0208224

[5] A. R. Blass, Near coherence of filters, I: Cofinal equivalence of models of arith-metic, Notre Dame Journal of Formal Logic 27 (1986), 579591.

[6] A. R. Blass, Near coherence of filters, II: Applications to operator ideals, theStone- Cech remainder of a half-line, order ideals of sequences, and slendernessof groups, Transactions of the American Mathematical Society 300 (1987),557580.

[7] A. R. Blass, Groupwise Density, Lecture given at the Mathematics Sym-posium in honor of Professor Saharon Shelah, Ben-Gurion University,May 21, 2001.

[8] A. R. Blass, Combinatorial cardinal characteristics of the continuum, preprint,to appear in the Handbook of Set Theory.

[9] A. R. Blass, Nearly adequate sets, Logic and Algebra (Yi Zhang, ed.), Con-temporary Mathematics 302 (2002), 3348.

[10] A. R. Blass and S. Shelah, There may be simple P1- andP2-points, and theRudin-Keisler ordering may be downward directed, Annals of Pure and AppliedLogic 33 (1987), 213243.

[11] A. R. Blass and S. Shelah, Near coherence of filters, III: A simplified consis-tency proof, Notre Dame Journal of Formal Logic 27 (1986), 579591.

[12] E. K. van Douwen, The integers and topology, in: Handbook of Set The-oretic Topology (K. Kunen and J. Vaughan, eds.), North-Holland, Amster-dam, 1984, 111167.

[13] F. Galvin and A. W. Miller, sets and other singular sets of reals, Topologyand its Applications 17 (1984), 145155.

[14] J. Gerlits and Zs. Nagy, Some properties of C(X), I, Topology and its appli-cations 14 (1982), 151161.

[15] W. Hurewicz, Uber eine Verallgemeinerung des Borelschen Theorems, Mathe-matische Zeitschrift 24 (1925), 401421.

[16] W. Hurewicz, Uber Folgen stetiger Funktionen, Fundamenta Mathematicae 9(1927), 193204.

[17] W. Just, A.W. Miller, M. Scheepers and P.J. Szeptycki, The combinatorics ofopen covers II, Topology and its Applications 73 (1996), 241266.

[18] R. Laver, On the consistency of Borels conjecture, Acta Mathematica 137(1976), 151169.

[19] M. K. Menger, Einige Uberdeckungssatze der Punktmengenlehre, Sitzungs-berichte der Wiener Akademie 133 (1924), 421444.

[20] I. Reclaw, Every Luzin set is undetermined in the point-open game, Funda-menta Mathematicae 144 (1994), 4354.

[21] F. Rothberger, Sur des families indenombrables de suites de nombres naturels,et les problemes concernant la propriete C, Proceedings of the CambridgePhilosophical Society 37 (1941), 109126.

[22] M. Sakai, PropertyC and function spaces, Proceedings of the American Math-ematical Society 104 (1988), 917919.


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[23] M. Scheepers, Combinatorics of open covers I: Ramsey Theory, Topology andits Applications 69 (1996), 3162.

[24] M. Scheepers, Sequential convergence inCp(X) and a covering property, East-West Journal of Mathematics 1 (1999), 207214.

[25] M. Scheepers, The length of some diagonalization games, Arch. Math. Logic38 (1999), 103122.

[26] M. Scheepers and B. Tsaban, The combinatorics of Borel covers, Topology andits Applications 121 (2002), 357382.http://arxiv.org/abs/math.GN/0302322

[27] S. Shelah and B. Tsaban, Critical cardinalities and additivity properties ofcombinatorial notions of smallness, Journal of Applied Analysis 9 (2003), 149162.http://arxiv.org/abs/math.LO/0304019

[28] B. Tsaban, A topological interpretation of t, Real Analysis Exchange 25(1999/2000), 391404.http://arxiv.org/abs/math.LO/9705209

[29] B. Tsaban, Selection principles and the minimal tower problem, Note diMatematica, to appear.http://arxiv.org/abs/math.LO/0105045

[30] B. Tsaban, A diagonalization property between Hurewicz and Menger, RealAnalysis Exchange 27 (2001/2002), 757763.http://arxiv.org/abs/math.GN/0106085

Department of Mathematics and Computer Science, Boise State Uni-

versity, Boise, Idaho 83725 U.S.A.

E-mail address: [email protected]: http://math.boisestate.edu/~tomek

Institute of Mathematics, Hebrew University of Jerusalem, Givat

Ram, 91904 Jerusalem, Israel, and Mathematics Department, Rutgers

University, New Brunswick, NJ 08903, U.S.A.

E-mail address: [email protected]

Department of Mathematics and Computer Science, Bar-Ilan Uni-

versity, Ramat-Gan 52900, Israel

E-mail address: [email protected]: http://www.cs.biu.ac.il/~tsaban

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Tomek Bartoszynski, Saharon Shelah and Boaz Tsaban- Additivity Properties of Topological Diagonalizations