# Andrzej Roslanowski and Saharon Shelah- Localizations of infinite subsets of omega

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Localizations of infinite subsets of

Andrzej Roslanowski

Institute of Mathematics

The Hebrew University of Jerusalem

Jerusalem, Israel

and

Mathematical Institute of Wroclaw University

50384 Wroclaw, Poland

Saharon Shelah

Institute of Mathematics

The Hebrew University of Jerusalem

Jerusalem, Israel

and

Department of Mathematics

Rutgers University

New Brunswick, NJ 08854, USA

done: September 1992printed: October 6, 2003

The research was partially supported by Polish Committee of Scientific Research, GrantKBN 654/2/91

The second author would like to thank Basic Research Foundation of The Israel Academyof Sciences and Humanities for partial support. Publication number 501.

0

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0 Introduction0.1 Preliminaries

When we say reals we mean one of the nicely defined Polish spaces or their(finite or countable) products like: the real line R, the Cantor space 2 , theBaire space or the space of infinite sets of integers [] .In the present paper we are interested in properties of forcing notions (or, gen-erally, extensions of models of ZFC) which measure in a sense the distancebetween the ground model reals and the reals in the extension. In particular welook at the ways the new reals can be localized (or: aproximated) by oldreals. There are two extreme cases here: there are no new reals and the old realsare countable. However, between these two extremes we have a wide spectrum

of properties among which the localizations by slaloms seem to be the mostpopular. A systematic study of slaloms and related localization properties andcardinal invariants was presented in [Bar1].

A slalom is a function S : []< such that (n)(|S(n)| = n + 1). Wesay that a slalom S localizes a function f whenever (n )(f(n) S(n)).In this situation we can think that the slalom S is an approximation of thefunction f. It does not determine the function but it provides some boundson possible values of f. Bartoszynski, Cichon, Kamburelis et al. studied thelocalization by slaloms and those investigations gave the following surprisingresult.

Theorem 0.1 (Bartoszynski, [Bar2]) Suppose that V V are models ofZFC. Then the following conditions are equivalent:

1. Any function from V can be localized by a slalom from V.

2. Any Borel (Lebesgue) null set coded in V can be covered by a Borel nullset coded in V.

On localizations by slaloms see Chapter VI of [Sh:b] too; other localizations ofslalom-like type appeared in [GoSh:448].A stronger localization property was considered in [NeRo]. Fix a natural numberk 2. By a k-tree on we mean a tree T

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Definition 0.2 Assume that X, Y are Polish spaces and R X Y is a Borelrelation. Suppose that V V are models of ZFC and that all parameterswe need are in V. We say that the pair (V, V) has the property of the R-localization if

(xXV)(y YV)((x, y) R)

If x X V, y Y V and (x, y) R they we say that y R-localizes x.

In the examples we gave earlier X was and Y was the space of slaloms orthe space of all k-trees, respectively. The respective relations should be obvious.Those localizations were to approximate functions in an extension by objectsfrom the ground model. They are not useful if we consider infinite subsets of .Though each member of [] can be identified with its increasing enumeration,the localization (either by slaloms or by k-trees) of the enumeration does not

provide satisfactory information on successive points of the set. The localizationgives us candidates for the n-th point of the set but the same candidates canappear several times for distinct n. That led to a suggestion that we shouldconsider disjoint subsets of as sets of candidates for successive points of thelocalized set (the approach was suggested by B. Weglorz). Now we have twopossibilities. Either we can demand that each set from the localization containsa limited number of members of the localized set or we can postulate that eachintersection of that kind is large. Localizations of this kind are studied in section1. In the second section we investigate localizations of infinite subsets of bysets of integers from the ground model. These localizations might be thoughtas localizations by partitions of into successive intervals. A starting point forour considerations was the following observation.

Proposition 0.3 Suppose that V V are (transitive) models of ZFC. Then:

1. V is unbounded in V if and only if

for every set X [] V there exists a set Y in[] V such thatinfinitely often between two successive points of Y there are at least 2points of X.

2. V is dominating in V if and only if

for every set X [] V there exists a set Y [] V such that forall but finitely many pairs of two successive points of Y there are atleast 2 points of X between them.

Now we try to replace the quantifier for infinitely manyabove by stronger quan-tifiers (but still weaker than for all but finitely many), like for infinitely manyn, for both n and n + 1. Finally, in section 3 we formulate several corollaries tothe results of previous sections for cardinal invariants related to the notions westudy.

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0.2 Notation

Our notation is rather standard and essentially compatible with that of [Jec]and [Kun]. In forcing considerations, however, we will use the convention thata stronger condition is the greater one.

Basic Definitions 0.4 1. A tree on is a setT < closed under initialsegments. For the tree T the body [T] of T is the set

{x : (l )(xl T)}.

If t T then succT(t) = {s T : t s & lh(t) + 1 = lh(s)}.

2. By a model of ZFC we will mean a transitive model of (enough of) ZFC.

Models of ZFC will be denoted by V, V

etc.3. We will be interested in extensions of models, i.e. in pairs (V, V) of

models such that V V. If a property of an extension is defined thenwe extend this definition to notions of forcing. We say that a notion offorcing P has the property whenever for any generic filter G P over Vthe extension V V[G] has the considered property.

4. We will use the quantifiers (n) and (n) as abbreviations for

(m )(n > m) and (m )(n > m),

respectively.

5. The Baire space of all functions from to is endowed with thepartial order :

f g (n)(f(n) g(n)).

A family F is unbounded in (, ) if

(g )(f F)(f g)

and it is dominating in (, ) if

(g )(f F)(g f).

6. The unbounded numberb is the minimal size of an unbounded family in(, ); the dominating number d is the minimal size of a dominatingfamily in that order.

7. The size of the continuum is denoted by c, [] stands for the family ofinfinite co-infinite subsets of .

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0.3 Acknowledgement

We would like to thank Professor Uri Abraham for his helpful comments on thepaper.

1 Rk, Rk - localizations

In this section we show that a localization of infinite subsets of suggested byB. Weglorz implies that the considered extension adds no new real.

Definition 1.1 1. A partition of into finite sets is a sequence Kn : n of disjoint finite sets such that

n Kn = .

2. Pk is the set of all partitions Kn : n of into finite sets such that(n )(|Kn| > k).

[Note that Pk is a 02-subset of ([]< ) so it is a Polish space.]

3. We define relations Rk [] Pk and Rk []

Pk+1 by

(X, Kn : n ) Rk (

n)(|X Kn| k)

(X, Kn : n ) Rk (

n)(|X Kn| k).

Their complements (in [] Pk, [] Pk+1) are denoted by cRk, cR

k,

respectively.

If we want to approximate an infinite co-infinite subset of by an object in

a given model we can look for a separation of distinct members of the set bya sequence of sets from the model. Thus we could ask if it is possible to finda partition of (in V) such that the localized set is a partial selector of thepartition. More generally we may ask for Rklocalization; recall definition 0.2.Thus the Rk-localization property means that for every infinite set of integers Xfrom the extension there exists a partition Kn : n Pk from the groundmodel such that for almost all n the intersection X Kn is of size at mostk. The following result shows that the Rk-localization fails if we add new reals.

Theorem 1.2 Suppose that V V are models of ZFC such that V 2 =V 2. Then there is a set X [] V such that for no k there is apartition Kn : n V of such that

(n )(|X Kn| k & |Kn| > k).

Consequently the extension V V does not have the Rk-localization property(for any k).

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