Alan Dow and Saharon Shelah- More on Tie-Points and Homeomorphism in N^*

8/3/2019 Alan Dow and Saharon Shelah- More on Tie-Points and Homeomorphism in N^*

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MORE ON TIE-POINTS AND HOMEOMORPHISM INN

ALAN DOW AND SAHARON SHELAH

Abstract. A point x is a (bow) tie-point of a space X if X \ { x}can be partitioned into (relatively) clopen sets each with x in itsclosure. We picture (and denote) this as X = A

xB where A, B

are the closed sets which have a unique common accumulationpoint x. Tie-points have appeared in the construction of non-trivial autohomeomorphisms of N \ N = N (e.g. [10, 7]) andin the recent study [4, 2] of (precisely) 2-to-1 maps on N . Inthese cases the tie-points have been the unique xed point of aninvolution on N . One application of the results in this paper is theconsistency of there being a 2-to-1 continuous image of N whichis not a homeomorph of N .

1. Introduction

A point x is a tie-point of a space X if there are closed sets A, B of X such that {x} = A B and x is an adherent point of both A and B.We let X = A x B denote this relation and say that x is a tie-point aswitnessed by A, B . Let A x B mean that there is a homeomorphismfrom A to B with x as a xed point. If X = A x B and A x B, thenthere is an involution F of X (i.e. F 2 = F ) such that {x} = x( F ). Inthis case we will say that x is a symmetric tie-point of X .

An autohomeomorphism F of N is said to be trivial if there is abijection f between conite subsets of N such that F = f N .Since the xed point set of a trivial autohomeomorphism is clopen, asymmetric tie-point gives rise to a non-trivial autohomeomorphism.

If A and B are arbitrary compact spaces, and if x A and y B

are accumulation points, then let A x = y B denote the quotient space of

Date : August 12, 2007.1991 Mathematics Subject Classication. 03A35.Key words and phrases. automorphism, Stone-Cech, xed points.Research of the rst author was supported by NSF grant No. NSF-. The

research of the second author was supported by The Israel Science Foundationfounded by the Israel Academy of Sciences and Humanities, and by NSF grant No.NSF- . This is paper number in the second authors personal listing.

1


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2 A. DOW AND S. SHELAH

A B obtained by identifying x and y and let xy denote the collapsedpoint. Clearly the point xy is a tie-point of this space.

In this paper we establish the following theorem.Theorem 1.1. It is consistent that N has symmetric tie-points x, ymainas witnessed by A, B and A , B respectively such that N is not home-omorphic to the space A x = y A

Corollary 1.1. It is consistent that there is a 2-to-1 image of N which is not a homeomorph of N .

One can generalize the notion of tie-point and, for a point x N ,consider how many disjoint clopen subsets of N \ { x} (each accumu-lating to x) can be found. Let us say that a tie-point x of N satises (x) n if N \ { x} can be partitioned into n many disjoint clopensubsets each accumulating to x. Naturally, we will let (x) = n denotethat (x) n and (x) n+1. Each point x of character 1 in Nis a symmetric tie-point and satises that (x) n for all n. We listseveral open questions in the nal section.

More generally one could study the symmetry group of a point xN : e.g. set Gx to be the set of autohomeomorphisms F of N thatsatisfy x(F ) = {x} and two are identied if they are the same on someclopen neighborhood of x.

Theorem 1.2. It is consistent that N has a tie-point x such that two

(x) = 2 and such that with N = A x B, neither A nor B is a homeomorph of N . In addition, there are no symmetric tie-points.

The following partial order P 2, was introduced by Velickovic in [10]to add a non-trivial automorphism of P (N)/ [N]< 0 while doing as littleelse as possible at least assuming PFA.

Denition 1.1. The partial order P 2 is dened to consist of all 1-to-1posetfunctions f : A B where

A and B for all i and n , f (i) (2n +1 \ 2n ) if and only if

i (2n +1 \ 2n ) lim sup n |(2n +1 \ 2n ) \ A| = and hence, by the previous

condition, lim sup n |(2n +1 \ 2n ) \ B | = The ordering on P 2 is .

We dene some trivial generalizations of P 2. We use the notationP 2 to signify that this poset introduces an involution of N becausethe conditions g = f f 1 satisfy that g2 = g. In the denitionof P 2 it is possible to suppress mention of A, B (which we do) and


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MORE ON TIE-POINTS AND HOMEOMORPHISM IN N 3

to have the poset P 2 consist simply of the functions g (and to treatA = min( g) = {i dom(g) : i < g (i)} and to treat B as max(g) ={i dom(g) : g(i) < i }).

Let P 1 denote the poset we get if we omit mention of f but consistingonly of disjoint pairs (A, B ), satisying the growth condition in Deni-tion 1.1, and extension is coordinatewise mod nite containment. Tobe consistent with the other two posets, we may instead represent theelements of P 1 as partial functions into 2.

More generally, let P be similar to P 2 except that we assume thatconditions consist of functions g satisfying that {i, g(i), g2(i), . . . , g (i)}has precisely elements for all i dom(g) (and replace the intervals2n +1 \ 2n by n +1 \ n in the denition).

The basic properties of P2

as dened by Velickovic and treated byShelah and Steprans are also true of P for all N.

In particular, for example, it is easily seen thatProposition 1.1. If L N and P = L P (with full supports) observeand G is a P -generic lter, then in V [G], for each L, there is a tie-point x N with (x ) .

For the proof of Theorem 1.1 we use P 2 P 2 and for the proof of Theorem 1.2 we use P 1.

An ideal I on N is said to be ccc over n [3], if for each uncountablealmost disjoint family, all but countably many of them are in I . An

ideal is a P -ideal if it is countably directed closed mod nite.The following main result is extracted from [6] and [8] which werecord without proof.Lemma 1.1 (PFA) . If P is a nite or countable product (repetitions mainlemmaallowed) of posets from the set {P : N} and if G is a P -generic lter, then in V [G] every autohomeomorphism of N has the property that the ideal of sets on which it is trivial is a P -ideal which is ccc over n.Corollary 1.2 (PFA) . If P is a nite or countable product of posets maincorollary from the set {P : N}, and if G is a P -generic lter, then in V [G] if

F is an autohomeomorphism of N and {Z : 2} is an increasing mod nite chain of innite subsets of N, there is an 0 2 and a collection {h : 2} of 1-to-1 functions such that dom(h ) = Z and for all 2 and a Z \ Z 0 , F [a] = h [a].

Each poset P as above is 1-closed and 2-distributive (see [8,p.4226]). In this paper we will restrict out study to nite products.The following partial order can be used to show that these productsare 2-distributive.


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Denition 1.2. Let P be a nite product of posets from {P : N}.Given { f : } = F P (decreasing in the ordering on P ), deneP (F ) to be the partial order consisting of all g P such that there issome such that g f . The ordering on P (F ) is coordinatewise

as opposed to in P .

Corollary 1.3 (PFA) . If P is a nite or countable product of posetsotherautos from the set {P : N}, and if G is a P -generic lter, then in V [G]if F is an involution of N with a unique xed point x, then x is a P 2 -point and N = A x B for some A, B such that F [A] = B .

Proof. We may assume that F also denotes an arbitrary lifting of F to [N] in the sense that for each Y N , (F [Y ]) = F [Y ]. Let

Z x = [N] \ x (the dual ideal to x). For each Z Z x , F [Z ] is alsoin Z and F [Z F [Z ]] = Z F [Z ]. So let us now assume that Z denotes those Z Z x such that Z = F [Z ]. Given Z Z , sincex(F ) Z = , there is a collection Y [Z ] such that F [Y ]Y =for each Y Y , and such that Z is covered by {Y : Y Y} . Bycompactness, we may assume that Y = {Y 0, . . . , Y n } is nite. SetZ 0 = Y 0 F [Y 0]. By induction, replace Y k by Y k \ j


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1.1, it follows that there is some 1 such that Z = C Z , F Z istrivial and for some uncountable I 1 {(Z a , Z b ) : I } formsa Hausdorff-gap. This however is a contradiction because if hZ inducesF Z , then min( hZ ) ((Z a ) (Z b )) is almost equal to a forall 1, i.e. min(hZ ) would have to split the Hausdorff-gap.

The forcing P (F ) introduces a tuple f which satises f f for f F but for the fact that f may not be a member of P simply

because the domains of the component functions are too big. Thereis a -centered poset which will choose an appropriate sequence f of subfunctions of f which is a member of P and which is still below eachmember of F (see [6, 2.1]).

A strategic choice of the sequence F will ensure that P (F ) is ccc,but remarkably even more is true. Again we are lifting results from[6, 2.6] and [8, proof of Thm. 3.1]. This is an innovative factoring of Velickovics original amoeba forcing poset and seems to preserve moreproperties. Let


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Lemma 1.3. Let F P (for any N) be generic over V , then for also-useful-lemma each P (F )-name h NN , either there is an f F such that f P (F )

h dom(f ) / V , or there is an f F and an increasing sequencen0 < n 1 < of integers such that for each i [n k , n k +1 ) and each g < f such that g forces a value on h(i), f (g [nk , n k +1 )) also forcesa value on h(i).

Proof. Given any f , perform a standard fusion (see [6, 2.4] or [8, 3.4])f k , n k by picking Lk [nk +1 , n k +2 ) (absorbed into dom( f k +1 )) so thatfor each partial function s on nk which extends f k nk , if there is someinteger i nk+1 for which no


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Let H be a generic lter for


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Given such a family as H , there is a well-known proper poset Q1 (see[1, 3.1], [3, 2.2.1], and [10, p9]) which will force an uncountable conalI and a collection of integers {k, : < I } satisfying thath (k, ) = h (k, ) (and both are dened) for < I . So, let Q1 bethe


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f < g (f, g P ) if g f . For each f P , let a f = f 1(0) andbf = f 1(1).

Again we assume that {a : 2} is the sequence of P -namessatisfying that N = A x B and A \ { x} = {a : 2}. Of courseby this we mean that for each f G, there are 2, a [N] , andf 1 G such that a f a a f 1 and f 1 P a = a .

Next we assume that, if A is homeomorphic to N , then F is a P -name of a homeomorphism from N to A and let z denote the pointin N which F sends to x. Also, let Z be the P -name of F 1[a ]and recall that N \ { z} = {Z : 2}. As above, we may alsoassume that for each 2, there is a P -name of a function h withdom(h ) = Z such that F [Z ] is induced by h .

Furthermore if (x) > 2, then one of A \ { x} or B \ { x} can bepartitioned into disjoint clopen non-compact sets. We may assumethat it is A \ { x} which can be so partitioned. Therefore there is somesequence {c : 2} of P -names such that for each < 2,c a and c a = c . In addition, for each < 2 there must bea 2 such that c \ a and a \ (c a ) are both innite.

Now assume that H is


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L1 L0 such that lim k L 1 |S 1k | = . Notice that each member of i0k isthe minimum element of S 1k . Again, we may extend f 1 and assume thatN \ dom(f 1) is equal to k L 1 S 1k . Suppose now we have some innite L j ,some f j , and for k L j , an increasing sequence {i0k , i1k , . . . , i

j 1k } S 0k .

Assume further that

S jk { ik : < j } = S 0k \ dom(f j )

and that lim k L j |S jk | = . For each k L j , let i

jk = min( S

jk \ { ik :

< j }). By a simple recursion of length 2 j , there is an f j +1 < f j suchthat, for each k L j , {ik : j } S 0k \ dom(f j +1 ) and for eachfunction s from {ik : j } into 2, the condition f j +1 s forces a valueon h(i j

k). Again nd L j +1 L j so that lim k L

j+1 |S

j +1

k| = (where

S j +1k = S 0k \ dom(f j +1 )) and extend f j +1 so that N \ dom(f j +1 ) is equalto k L j +1 S

j +1k .

We are half-way there. At the end of this fusion, the function f = j f j is a member of P because for each j and k L j +1 , 2

m k +1 \ (2m k

dom(f )) { i0k , . . . , i jk }. For each k, let S k = S 0k \ dom(f ) and, by

possibly extending f , we may again assume that there is some L suchthat lim k L |S k | = and that, for k L, S k = {i0, i1k , . . . , i

j kk } for

some j k . What we have proven about f is that it satises that for eachk L and each j < j k and each function s from {i0k , . . . , i

j 1k } to 2,

f s (i

j

k , 0) forces a value onh(i

j

k ).To nish, simply repeat the process except this time choose maximalvalues and work down the values in S k . Again, by genericity of F , theremust be such a condition as f in F .

Returning to the proof of Theorem 3.1, we are ready to use Lemma3.1 to show that forcing with P (F ) will not introduce undesirable func-tions h analogous to the argument in Theorem 1.1. Indeed, assumethat we are in the case that F is a homeomorphism from N to A asabove, and that {h : } is the family of functions as above. If we show that h does not satisfy that h h for each , then we

proceed just as in Theorem 1.1. By Lemma 3.1, we have the conditionf 0 F and the sequence S k (k N) such that N \ dom(f 0) = k S kand that for each i k S k , f 0 { (i, 0)} forces a value (call it h(i)) onh(i). Therefore, h is a function with domain k S k in V . It suffices tond a condition in P below f 0 which forces that there is some suchthat h is not extended by h. It is useful to note that if Y k S kis such that lim sup |S k \ Y | is innite, then for any function g 2Y ,f 0 g P .


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We rst check that h is 1-to-1 on a conite subset. If not, there isan innite set of pairs E j k S k , h[E j ] is a singleton and such thatfor each k, S k j E j has at most two elements. If g is the functionwith dom( g) = j E j which is constantly 0, then f 0 g forces that hagrees with h on dom(g) and so is not 1-to-1. On the other hand, thiscontradicts that there is f 1 < f 0 g such that for some 2, aalmost contains ( f 0 g) 1(0) and the 1-to-1 function h with domaina is supposed to also agree with h on dom(g).

But now that we know that h is 1-to-1 we may choose any f 1 Fsuch that f 1 < f 0 and such that there is an 2 with f

10 (0)

a f 11 (0), and f 1 has decided the function h . Let Y be any innite

subset of N \ dom(f 1) which meets each S k in at most a single point. If h[Y ] meets Z in an innite set, then choose f 2 < f 1 so that f 2[Y ] = 0and there is a > such that Y a . In this case we will have thatf 2 forces that Y a \ a , h Y h , and h [Y ] h [a ] is innite(contradicting that h is 1-to-1). Therefore we must have that h[Y ] isalmost disjoint from Z . Instead consider f 2 < f 1 so that f 2[Y ] = 1.By extending f 2 we may assume that there is a < 2 such thatf 2 P (F ) Z h[Y ] is innite. However, since f 2 P (F ) h h, wealso have that f 2 P (F ) h (a \ a ) h and (a \ a ) Y = contradicting that h is 1-to-1 on dom(f 2) \ dom(f 0). This nishes theproof that there is no P (F ) name of a function extending all the h s

( ) and the proof that F can not exist continues as in Theorem1.1.Next assume that we have a family {c : } as described above

and suppose that C = h 1(0) satises that (it is forced) C a = cfor all . If we can show there is no such h, then we will knowthat in the extension obtained by forcing with P (F ), the collection{(c , (a \ c )) : } forms an (1, 1)-gap and we can use a properposet Q1 to freeze the gap. Again, meeting 1 dense subsets of theiteration


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Z , then f 1 forces that h[a \ dom(f 1)] = 0, and so (a \ dom(f 1)) C for all 2. However, taking so large that each of c \ dom(f 1)and (a \ (c dom(f 1)) are innite shows that no such h exists.

Finally we show that there are no involutions on N which have aunique xed point. Assume that F is such an involution and that zis the unique xed point of F . Applying Corollaries 1.2 and 1.3, wemay assume that N \ { z} = 2 Z and that for each , F Z isinduced by an involution h .

Again let H be


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Remark 1. The tie-point x3 introduced by P 3 does not satisfy that (x3) = 3. This can be seen as follows. For each f P 3, we canpartition min( f ) into {i dom(f ) : i < f (i) < f 2(i)} and {idom(f ) : i < f 2(i) < f (i)}.

It seems then that the tie-points x introduced by P might be bettercharacterized by the property that there is an autohomeomorphism F of N satisfying that x( F ) = {x }, and each y N \ { x} has an orbitof size .

Remark 2. A small modication to the poset P 2 will result in a tie-pointN = A x B such that A (hence the quotient space by the associatedinvolution) is homeomorphic to N . The modication is to build into

the conditions a map from the pairs {i, f (i)} into N. A natural way todo this is the poset f P +2 if f is a 2-to-1 function such that for eachn, f maps dom( f ) (2n +1 \ 2n ) into 2n \ 2n 1, and again lim sup n |2n +1 \(dom( f ) 2n )| = . P +2 is ordered by almost containment. The genericlter introduces an 2-sequence {f : 2} and two ultralters:x { N \ dom(f ) : 2} and z { N \ range( f ) : 2}. Foreach and a = min( f ) = {i dom(f ) : i = min( f 1 (f (i))}, weset A = {x} a and B = {x} (dom( f ) \ a ) , and we havethat N = A x B is a symmetric tie-point. Finally, we have thatF : A N dened by F (x) = z and F A \ { x} = (f ) is ahomeomorphism.

Question 4.2. Assume PFA. If L is a nite subset of N and P L ={P : L}, is it true that in V [G] that if x is tie-point, then (x) L; and if 1 / L, then every tie-point is a symmetric tie-point?

References

[1] Alan Dow, Petr Simon, and Jerry E. Vaughan, Strong homology and the proper forcing axiom , Proc. Amer. Math. Soc. 106 (1989), no. 3, 821828. MRMR961403 (90a: 55019)

[2] Alan Dow and Geta Techanie, Two-to-one continuous images of N , Fund.Math. 186 (2005), no. 2, 177192. MR MR2162384 (2006f: 54003)

[3] Ilijas Farah, Analytic quotients: theory of liftings for quotients over analyticideals on the integers , Mem. Amer. Math. Soc. 148 (2000), no. 702, xvi+177.MR MR1711328 (2001c: 03076)

[4] Ronnie Levy, The weight of certain images of , Topology Appl. 153 (2006),no. 13, 22722277. MR MR2238730 (2007e: 54034)

[5] S. Shelah and J. Stepr ans. Non-trivial homeomorphisms of N \ N without theContinuum Hypothesis. Fund. Math. , 132:135141, 1989.

[6] S. Shelah and J. Stepr ans. Somewhere trivial autohomeomorphisms. J. London Math. Soc. (2) , 49:569580, 1994.


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[7] Saharon Shelah and Juris Stepr ans, Martins axiom is consistent with the ex-istence of nowhere trivial automorphisms , Proc. Amer. Math. Soc. 130 (2002),

no. 7, 20972106 (electronic). MR 1896046 (2003k: 03063)[8] Juris Stepr ans, The autohomeomorphism group of the Cech-Stone compacti-cation of the integers , Trans. Amer. Math. Soc. 355 (2003), no. 10, 42234240(electronic). MR 1990584 (2004e: 03087)

[9] B. Velickovic. Denable automorphisms of P ()/fin . Proc. Amer. Math. Soc. ,96:130135, 1986.

[10] Boban Velickovic. OCA and automorphisms of P ()/ n. Topology Appl. ,49(1):113, 1993.

Department of Mathematics, Rutgers University, Hill Center, Pis-cataway, New Jersey, U.S.A. 08854-8019

Current address : Institute of Mathematics, Hebrew University, Givat Ram,Jerusalem 91904, Israel

E-mail address : [email protected]

Alan Dow and Saharon Shelah- More on Tie-Points and Homeomorphism in N^*

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