reflection and weakly collectionwise hausdorff spaces

PROCEEDINGS OF THEAMERICAN MATHEMATICAL SOCIETYVolume 122, Number 1, September 1994

REFLECTION AND WEAKLY COLLECTIONWISEHAUSDORFF SPACES

TIM LABERGE AND AVNER LANDVER

(Communicated by Franklin D. Tall)

Abstract. We show that 0(0) implies that there is a first countable < 6-

collectionwise Hausdorff space that is not weakly (9-collectionwise Hausdorff.

We also show that in the model obtained by Levy collapsing a weakly compact

(supercompact) cardinal to a>2, first countable ¡»^-collectionwise Hausdorff

spaces are weakly ^-collectionwise Hausdorff (weakly collectionwise Haus-

dorff). In the last section we show that assuming E% , a certain 0-family of

integer-valued functions exists and that in the model obtained by Levy collaps-

ing a supercompact cardinal to u>i . these families do not exist.

1. Introduction

Reflection is a central theme in modern set-theoretic topology. As Alan Dow

points out in [Do], we often prove theorems when some type of reflection prin-

ciple holds, and we build counterexamples when reflection fails. This paper

contains both types of results, on questions related to the failure of collection-

wise Hausdorff.We say that a subset A of a topological space X can be separated if there is

a collection {Ux : x e A} of disjoint open sets with x eUx for every x e A.

A space X is < 9-collectionwise Hausdorff ( < 0-cwH) if every closed discrete

subset of size < 6 can be separated. A space X is < 6-collectionwise Hausdorff( < 0-cwH) if every closed discrete subset of size < 6 can be separated. X is

collectionwise Hausdorff (cwH) if it is < (9-cwH for every cardinal 8 .

In particular, we are concerned with Fleissner's questions:

Question 1. Is there a ZFC example of a first countable < #2-cwH space that isnot <v\2-cwW.

Question 2. Is there a ZFC example of a first countable < H2-cwH space that isnot cwffl

Fleissner asks for ZFC examples because he showed [F] that E%2 (a non-

reflecting stationary subset of {a < co2 : cf(a) = co} ) can be used to construct

an example of a first countable < N2-cwH space that is not < N2-cwH Since the

Received by the editors December 4, 1992.

1991 Mathematics Subject Classification. Primary 54D15; Secondary 54A35, 04A72.Key words and phrases. Reflection, weakly collectionwise Hausdorff, Levy forcing, Mitchell forc-

ing, fans.

© 1994 American Mathematical Society

0002-9939/94 $1.00 + $.25 per page

291

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

292 TIM LABERGE AND AVNER LANDVER

failure of E% is equiconsistent with the existence of a Mahlo cardinal [De, HS],

large cardinals are required for a negative answer to Question 1. In fact, we will

see that by a result of Todorcevic [To2], at least a weakly compact cardinal willbe needed to get a negative answer to Question 1. Shelah has shown [S] that, inthe model obtained by Levy collapsing a weakly compact cardinal to co2 , first

countable <N2-cwH spaces that are locally of size at most Ni are <N2-cwH, and

that in the model obtained by collapsing a supercompact cardinal, such spaces

are cwH. For a more complete history of the general problem of reflecting the

failure of cwH, see [W, FS]. We thank Bill Fleissner for his valuable input and

Gary Gruenhage for letting us use the handwritten notes [GT].

2. Squares of fans and first countablecollectionwise hausdorff spaces

Recently, Gruenhage and Tamaño [GT] have discovered a connection be-

tween Fleissner's questions and the problem of determining the tightness of

the squares of certain fans. Before we discuss this connection, we make some

definitions.The fan Fe<ü) is the quotient space obtained by identifying the nonisolated

points of 0-many copies of the convergent co-sequence. To be precise, FB(Ú =

{*} U (8 x co), topologized so that points of 8 x co are isolated and so that an

open base at * is the family of all

Bf = {*}u{ia,n):n>f(a)} (feeco).

When working with the square of Fg m , one can always replace two functions

fx, f2eeco with g = max (/i, f2), so we use the family of all

Vg = Bg x Bg (g e eco)

as an open base for (*, *) in Fßttox Fg¡0).

We say that a set S C Fe¡co x Fe¡a) is 8-good if (*, *) € S, but for all

T e [S]<e , (*, *) £ T. Thus, "there is a 8-good set in FgtCa x Fgt(0" means

that the tightness of Fe w x Fe m is 8 and that this tightness is actually attained.

Finally, we say that a set H e [co x co]<co is closed downward (c.d.w.) if

whenever (n, m) e H, then n' < n and m' < m implies («', m!) e H.The Gruenhage-Tamano result is the equivalence (1) ^=> (2) in Theorem 1.

We have added the combinatorial equivalence (3) and the superficially stronger

topological characterization (4).

Theorem 1. Let 8 be an uncountable cardinal. TFAE:

( 1 ) There exists a 8-good subset of F6(ax Few.

(2) There exists a space X that is first countable, <8-cwH and not <8-cwH.(3) There exists %A = {Haß :a < ß < 8} ç[cox co]<w with each Haß c.d.w.

such that:

(a) for every A e [8]<e there is a function f : 8 —> co such that for all a < ß

in A, ifia),fiß)) i Haß;(b) for every f : 8 -> co there are a< ß < 8 such that (f(a), /(/?)) € Haß .

(4) There exists a space X that is first countable, zero-dimensional, < 8-cwH

and not < 8-cwH.

In order to put Theorem 1 in perspective, we need the definition of a difficult-

to-deny combinatorial principle. For 8 an uncountable regular cardinal, D(0)


REFLECTION AND WEAKLY COLLECTIONWISE HAUSDORFF SPACES 293

is the assertion that there is a family {Ca : a < 8} satisfying the following

conditions:

(i) Ca Ca is a club subset of a.

(ii) If a is a limit point of Cß , then Ca = Cß n a.(iii) There is no club C c 8 such that for every limit point a of C,

Ca = C n a.

0(8) is true for every regular 8 which is not weakly compact in L (see [Toi]).

Todorcevic showed (see [To2, Be]) that 0(8) can be used to construct a

(9-good set. Also, combining Theorem 1 with Fleissner's construction of a first

countable, < 0-cwH, not < 0-cwH space from Ef, we see that E™ can be usedto construct a 0-good set.

Because the proof of Theorem 1 is essentially the same as the proof of The-

orem 2, we only prove Theorem 2. Before the statement of the theorem we

define a weakening of collectionwise Hausdorff introduced by Tall [Tal]. We

say that a subset A of a space X is weakly separated if it has a subset of size

\A\ that is separated. X is weakly 8-cwH if every closed discrete subset ofsize 8 is weakly separated. X is weakly cwH if it is weakly 0-cwH for every

cardinal 8.

Theorem 2. Let k < 8 be uncountable cardinals. TFAE:

(1) There exists a space X that is first countable, <k-cwH, and not weakly

8-cwH.(2) There exists %* = {Haß : a < ß < 8} c [co x co]<(0, with each Haß

c.d.w., such that:

(a) for every A e [8]<K there is a function f : 8 —> co such that for all

a<ß in A, ifia),fiß)) i Haß ;(b) for every B e [8]e and every f ' : 8 —> co there are a < ß in B such

that ifia),fiß))eHaß.

(3) There exists a space X that is first countable, zero-dimensional, <k-cwH,

and not weakly 8-cwH.

Proof. (1) -^ (2) : Let 8 c X be a subset which is not weakly separated. For

every a e 8 let {U„{a) : n e co} be a decreasing neighbourhood base at a.

Lex

S = {((a,n),(ß, m)) : Un(a) n Um(ß) * 0}.

S isa subset of Fgtü) xf9(¡). For every B c 8, let SB = {((a, n), (ß, m)) eS : a, ß e B} . We claim that S satisfies the following two conditions

(i) For every A e [8]<K , (*,*) i_SA.

(ii) For every B e [8]e , (*,*)eSB.

To show (i), let A e [8]<K . Let / : A -> co be such that {Uf{a)(a) : a e A}is a separation of A in X. Let Vf be the open neighbourhood of (*, *) in

Fe,o) x Fe,w that is determined by f. It is easy to check that Vf n SA = 0.

For (ii), let B e [8]e . Lex f : 8 -> co, and let Vf be as above. Now, f\B isnot a code for a separation of B in X ; therefore, there are a < ß in B such

that Uf(a)(a) n Uf(ß)(ß) * 0 . But then ((a, f(a)), (ß, f(ß))) eSBnVf.Using the fact that (*, *) is not in the closure of any countable subset of S,

it is not hard to verify the following facts.



Fact 1. For every a < 8 there is h(a) e co such that for every n > h(a) and

for every ß < 8 the set {m e co:((a, n), (ß, m)) e S} is finite.

Fact 2. For every ß < 8 there is g(ß) e co such that for every m > g(ß) and

for every a < 8 the set {n e co : ((a, n), (ß, m)) e S} is finite.

Let T = S n (Bn x Bg). Clearly, T satisfies (i) and (ii). Now for every

a< ß <8 define

H'aß = {(n,m)ecoxco: ((a, n),(ß, m)) e T}.

Let Haß be the downward closure of H'~. Let us show that the Haß 's are

finite. It is enough to show that the H'aß 's are finite. Assume that this is false,

and let a < ß < 8 be such that H'aB is infinite. It follows that for every k e co

there are n, m > k such that (n, m) e H'aß . This implies that, (*, *) is in

the closure of {((a, n), (ß, m)) : (n,m) e H'aß}; therefore (*, *) is in the

closure of a countable subset of T, a contradiction.

Finally, it is not hard to see that {Haß : a < ß < 8} satisfies (a) and (b) of

(2) precisely because T satisfies (i) and (ii).

(2)^(3): As in [GT], first let / = {((<*, n), (ß, m)) : a < ß < 8A(n, m) eHaß} . Then let X = I u 8. Points in / are isolated, and for every y e 8 abase at y is given by

Uk(y) = {y}ü{((a,n),(ß,m))el:

(a = yAn>k)v(ß = yAm>k)} (k e co).

Clearly, X is first countable. To see that X is < /c-cwH let A e [8]<K .

Let / : 8 —> co be given by (2)(a). Then check that {[//(„)(a) : a e A} is

a separation of A. To see that X is not weakly 0-cwH let B e [8]e and

/ : 8 -> co. By (2)(b), there are a < ß in B such that (f(a), f(ß)) e Haß .

So

«a, f(a)),(ß,f(ß))) e Uf{a)(a)nUf(ß)(ß);

therefore, / does not code a separation of B .Finally, the finiteness of the Haß 's implies that each U/C(a) is clopen, and

therefore X is zero-dimensional.

(3) =>(1): Trivial, d

3. Independence results

In this section, we first show that D(0) implies that there is a first countable,

< 0-cwH space that is not weakly 0-cwH. We then demonstrate the consistency

of "first countable < K2-cwH spaces are weakly N2-cwH (weakly cwH)", assum-

ing the consistency of "there is a weakly compact (supercompact) cardinal".Since D(0) is true unless 0 is weakly compact in L, this gives that "first

countable < !<2-cwH spaces are weakly ^2-cwH" is equiconsistent with "there

is a weakly compact cardinal" and that the consistency of "first countable < N2-

cwH spaces are weakly cwH" implies the existence of an inner model with many

measurable cardinals [KM].

Theorem 3. Let 8 be a regular uncountable cardinal, and suppose that D(0)

holds. Then there is a zero-dimensional, first countable, < 8-cwH space that isnot weakly 8-cwH.



Proof. In [Be], Todorcevic constructs, from the assumption of D(0), a functionp2 that maps pairs a < ß < 8 into co. In §4 we will study integer-valued

families of functions, so to keep our notation consistent, we define for each

ß < 8 a function hß : ß -> co by

hß(a) = p2(a,ß) (a<ß).

By the properties of p2 cited in [Be], the family {hß : ß < 8} satisfies

(1) For all a < ß < 8, there is an naß e co such that for all Ç < a,

ha(í) < hß(C) + naß. (This is monotonicity [DW].)

(2) For all B e [8]e and all new, there exist a < ß in B such that

hß(a) > n.

For every a < ß < 8, define Haß = {(n, m) : n + m < hßia)} . Let us show

that %A = {Haß : a < ß < 8} satisfies (2) of Theorem 2 with k = 8 .For (2)(a) let A e [8]<K . Let y < 8 be a bound for A, and define f : 8-> co

byf(a) = {h?^ + n<xy i{a<y>

\ 0 otherwise.

Now let a < ß in A, then f(a) + f(ß) > hyia) + nßy > hßia). Therefore,

if{a),fiß)) iHaß.For (2)(b) let B e [8]e and / : 0 -» co. Then there exist B' e [B]e and

n e co such that for every a e B', fia) = n . But then there are a < ß in B'

such that hßia) > 2n . Hence f(a) + f(ß) = 2n < hßia), so (/(a), /(/?)) e

Haß- O

Since the hß 's also satisfy

(3) whenever A, B e [8]e and n e co, there are a < ß with a e A and

ß e B such that hßia) > n ,

it follows that the space constructed above is badly nonnormal.In [Ta2], Tall asks if GCH is consistent with the existence of a first count-

able, weakly Ni-cwH space that is not weakly fcVcwH. He asked for a consistent

with GCH example because Daniels has constructed [D] such a space fromMA + -*CH. Our example answers Tail's question positively, is < ^2-cwH, and

exists unless there is a weakly compact cardinal in L.The next two theorems will use the method of forcing 4- reflection. Good

references for this technique are [DTW1] and [DTW2]. We first prove a preser-

vation lemma, whose proof is motivated by Lemma 3.13 in [Be].

Main Lemma. Assume that P is an cox-closed partial order and that 8 > co2

is a regular cardinal. Assume that %f = {Hafi : a < ß < 8} c [co x co]<0) arec.d.w., and the following condition from Theorem 2 holds:

(b) For every B e [8]e and every f : 8 -» co there are a < ß in B such

that (fia),f(ß))eHaß.

Assume that G is a f-generic filter over V. Then in V[G], for every f : 8 -» co

there are a < ß < 8 such that (fia), fiß)) e Haß .

Proof. Assume that the lemma is false. Then there is po e P and a P-name g

such that

Po \r "g : 8 -- CO A (Va < ß < 8) (g(a), g(ß)) i Haßn.



Let M be an elementary substructure of some H (A) ( A large enough) with

0 , MA, P, po, g, and lhP in M and M n 0 = ß < 8 .If possible, choose px < p0 and a0 < ß such that px e M and px II-

"(£(a0), 0) e Haoßn. Note that this would imply that px lh "giß) > 0".

Similarly construct Po > Pi > • • • > pn+\ • • • such that pn+x e M and an < ßwith

Pn+l\\-uigian),n)eHanß".

In particular, this would imply that pn+x lh "giß) > ri". This process must

stop at a finite stage since otherwise one could choose q with q <p„ for every

n e co and get that for every neco, q\V"giß)> ri", which is impossible.So, there is n e co such that for every q < pn and for every a < ß, if

q e M, thenqVr-«(g(a),n)eHaß".

Claim 1. For every a < ß , pH lh "(¿(a), ») £ 7/a/)".

Proof. Assume not. Then there is q < pn and a < ß such that q lh

"(g(a), «) e /iQ/'. Fix such a < ß and the corresponding Haß . Then //(A) |=

(3«? < Pn) Q lh "(<?(<*), n) e Haß". But p-n , lhP, ¿, a, and fl^ are all in M ;therefore,

M\=3q<pn(q\\-"(g(a),n)eHaß"),

which is a contradiction. D

Let B = {£ e 0 : Va < ß (pn lh "(¿(a), ») £ fl^")} • By elementarily of

M, \B\ = 8.Now, for every a e 8 let qa < pn and ma e co be such that <7a lh "^(a) =

mQ". In V, define a function /:ö->cu by

/(a) = ma + ñ (a € 0).

The following claim will give us the desired contradiction.

Claim 2. For every ß eB and every a < ß , (f(a), fiß)) i Haß .

Proof. Fix ß e B and a < ß. By definition, /(a) > ma and /(yS) > ñ,

therefore it is enough to show that ima, ñ) £ Haß . But pn I h "(£(a), n) $

Haßn, and qa < pn , and qa lh "¿(a) = mQ". Therefore qa lh "(wQ, ft) i Haß",

and this implies that indeed ima, n) £ Haß . D

Let K be strongly inaccessible and let P = PK be the Levy collapse of ?c to

co2 with countable conditions (for a definition and proofs of the facts below,

see [K]). Let G be a P-generic filter. We will use the following facts about

P: P is an cox-closed partial order. P is k-c.c. and k = co2 in V[G]. For

every A < k , P can be factored as P = PA x P^ , P n Vx = PA and in V[Gf],

P1 is forcing equivalent to P (in particular, it is cox-closed). For a definition

and discussion of weakly compact cardinals, see [K, KM]. The following typeof proof is well known [Ba, Mi]; see also [DTW2].

Theorem 4. Assume that k is a weakly compact cardinal. Let P be the Levy

collapse of k to co2 with countable conditions, and let G be a F-generic filter

over V. Then in V[G], every first countable <W2-cwH space is weakly W2-cwH.

Proof. By way of contradiction, suppose that in V[G] there is a first countable

< K2-cwH space that is not weakly ^2-cwH. In V[G], co2 = k; therefore, there

is %A = {Haß : a < ß < k} that satisfies (2) of Theorem 2 with k = 8. This



fact must be forced, and the forcing statement is nj over (VK , e) (with few

extra parameters). Using the facts that k is weakly compact (IlJ-reflection)

and that P is k-cc, we find an inaccessible A<k. such that in V[Gf], {Haß :a < ß < A} satisfies (2) of Theorem 2 (with A playing the role of both k and0).

Now V[Gf] \= P1 is «i-closed. Therefore, by the Main Lemma, for everyfunction f : A -> co in V[G], there are a < ß < A such that (/(a), /(/?)) eHaß . But A < k ; therefore, this contradicts the fact that (2)(a) of Theorem 2holds for JT in V[G]. D

For a definition and discussion of supercompact cardinals see [KM, J,DTW1].

Theorem 5. Assume that k is a supercompact cardinal and that the GCH holdsabove k . Let P be the Levy collapse of k to co2 with countable conditions, and

let G be a f-generic filter over V. Then in V[G] every first countable < #2-cwH

space is weakly cwH.

Proof. We have to show that in V[G], if 0 > co2 is a cardinal and A' is a first

countable, < N2-cwH space, then X is weakly 0-cwH. It is a consequence of aresult of Watson [W] that if 0 is a singular strong limit cardinal and X is first

countable and weakly < 0-cwH, then X is weakly 0-cwH. Therefore, we may

assume that 0 is regular. Since k = co2 in V[G], we have 8 >k . The plan is

to show that (2) of Theorem 2, with k = co2, fails in V[G].Let j:V ^ M be an elementary embedding such that:

(i) j(a) = a, for all a < k ,

(ii) j(K)>6,(iii) eMnVcM.

Now, P c VK and j \ VK is the identity map; therefore, P C M andby (iii) P e M. So M[G] makes sense. In M (and also in V), j(V) isthe Levy collapse of j(k) to co2. In the usual way j(V) can be factored, so

j(P) = j(P)K x ji¥)K . But JiF)K = P, so ji¥) = P x ;'(P)K . Moreover,

M[G] \= JiV)K is an cox -closed partial order.

Let / be a y(P)K-generic filter over V[G]. I is also y(P)'c-generic over

M[G], and there is a filter K c ;'(P) such that M[G][I] = M[K] and V[G][I] =V[K]. Using the fact that for every p G P, jip) = p, we can define, in V[K],

/ : V[G] - M[K]

that extends j and show that it is an elementary embedding (see [KM, §25]).

Let %A e V[G] with ¿T = {Haß : a < ß < 8} c [co x co]<w c.d.w., and

assume that the following condition from Theorem 2 holds in V[G] :

(b) For every B e [8]e and every / : 0 -► co there are a < ß in B such

that ifia),fiß))eHaß.

We will show that (a) from Theorem 2 (with k = co2 ) is false for £? in V[G].

%? has a name of size 0 in V that is a subset of M ; therefore, by (iii)

this name is in M. Hence, %* e M[G] and it is clear that condition, (b)holds for X in M[G] as well. Notice that formally ^ is a function givenby ;r = {((a, ß),Haß) : a < ß < 8}. Let &' = j*(%T) e M[K]. Then%" - {((y, à), H's) : y < ó < j(8)}, where each H's is a c.d.w. subset of



[co x co]«0. Notice that for every a < ß < 8, Haß = JiHaß) = H'j(a)j(ß).

Let A = j"8, then j"MT = j*"XA = {((j(a), j(ß)), Haß) : a < ß < 8} ={((y, ô),H'y6) : (y < ô) A (y, ô e A)} c %". The map j \ %T is a subset

of M of size 0, but it is not in V ; therefore, we cannot conclude that itis in M. Fortunately, 6M[G] n V[G] c M[G] (see [J, p. 463]); therefore,; r ¿T e M[G] c M[K].

All that implies that the following two statements are equivalent in M[K] :

(*) For every / : j(8) —> co there are y < 5 in A such that (f(y), f(ó)) e

Hys-(**) For every / : 0 —> co there are a < ß < 8 such that (f(a), f(ß)) e

Haß ■

As was metioned before, (b) holds for & in M[G] and M[G] \= "j(V)K isan CfJi-closed partial order". Therefore, by the main lemma, (**) (and hence

(*) ) holds in M[K].We conclude that in M[K], there is A e [Jid)]<jW such that for every

/ : j{6) - co there are y < ô in A with (/(y), f(ô)) e j*i^)yS (= H'y5).

Therefore, by elementarily of /*, in V[G] there is an A e [0]-Nl such that

for every f : 8 -> co there are a< ß in A with (/(a), fiß)) 6 Haß . O

We can also prove Theorems 4 and 5 replacing the Levy collapse by the

Mitchell collapse [Mi, DJW] and thus obtain the conclusions of these theoremsin models where CH fails.

From the existence of a huge cardinal, Tall [Ta2] obtains the consistency of"first countable weakly N»-cwH spaces are weakly ^2-cwH". He uses an even

stronger axiom of infinity to obtain the consistency of "first countable weakly

Ni-cwH spaces are weakly cwH". So we use much weaker hypotheses to obtainslightly weaker results.

4. Integer-valued functions

As was mentioned before, Todorcevic proved that D(0) implies that there is

a 0-good subset of FßtuJxFgtW [To2]. In [DW], it is proved that all one needsis a nonextendible monotone 0-family of functions (which exists under D(0)

but not in the model of Theorem 5).

Let g, h be functions with range a subset of co. We say that g weakly

bounds h if there is n e co such that for every x e dom(g) n dom(A)

gix) + n> hix) (in short, g + n > h).

We say that %? = {hß : ß < 8} is a 0-family of functions if the domain

of hß is ß. The 0-family %? is weakly bounded if there is a g that weaklybounds each element of %?. We say that %A is an initially weakly bounded(i.w.b.) 8-family if for every A e [8]<e , {hß : ß e A} is weakly bounded. Thefamily is nonextendible if {hß : ß < 0} is not weakly bounded.

We can now repeat the construction of a 0-good subset of Fe<(a x Fgt0)from a nonextendible i.w.b. 0-family {hß : ß < 8}, using i.w.b. in place of

monotonicity. To see this, define for every a < ß < 8 the finite set Haß =

{(n, m) e co x co : n + m < hß(a)} and check that <%* = {Haß : a < ß < 8}

satisfies (3) of Theorem 1.



Question 3. Does the existence of a 8-good subset of Fe>co x Fe<ú) imply theexistence of a nonextendible i.w.b. 8-family of functions^.

Fleissner [F] used the combinatorial principle E% to construct a locallycountable, locally compact Moore space that is < 0-cwH but not < 0-cwH.

In particular, by [GT] (see (1) •$=-=>■ (2) of Theorem 1), E% implies the

existence of a 0-good subset of Fe>0) x FStW.

Theorem 6. Let 8 be an uncountable regular cardinal, and let E c {a e 6 :

cofia) = co} be a nonreflecting stationary set. Then there is a 8-family {hß :

ß < 8} such that for every A c 0, {hß : ß e A} is weakly bounded if and onlyif AC\E is nonstationary. (/« particular, {hß : ß < 8} is a nonextendible i.w.b.

8-family.)

Proof. For every a e E, fix {aa(«) : n e co} an increasing «-sequence un-bounded in a. For every a < ß < 0 define

min{n e co : ûa(n) ¿ ctß{n)} if iaeE)AißeE),

0 otherwise.hßia) = I

It is enough to consider Ac E. Let A be stationary, and assume, by way of

contradiction, that g : 8 —> co is a weak bound for {hß : ß e A} . There existsa stationary B c A and n e co such that for every ß e B, g + n > hß . Letf = g + n; then for every ß e B, f> hß .

Now, there exists a stationary S C B, and m e co such that for every

a,ß e S, fia) = m and (aa(0), ..., a„(m)) = (aß(0),..., aß(m)). Let

a < ß in S. Let k = min{n e co : aa(n) ^ ap(n)}. Then k > m; therefore,/(a) = m < k = hß(a), a contradiction.

Next we prove, by induction on y < 8, that there exists gy : y —> co thatweakly bounds {hß : ß < y} .

The successor case is easy, so let us assume that y is a limit. Let C c y bea club with C n E n y = 0. For every ß e En y let

y+(ß) = mm(C\ß), y~(ß) = max(Cnß).

Notice that for every ß e E Dy we have y~(ß) < ß < y+(ß) < y. Nowdefine

g7V)(ß) ifßtE,

otherwise.

Let us show that gy weakly bounds {hß : ß < y}. If/? <£ E, then hß isidentically zero, so it suffices to consider ß e Er\y. Let n = k+m+1, where kis the least integer such that üß(k) > y~(ß) and m is satisfies gy^) + m > hß .

Let us show that gy + n> hß . Let a e E n ß .

Case 1: y~(a) = y~(ß). In this case, y+(a) = y+(ß); therefore gy(a) =

gy^a)(a) = gy^ß)(a). But gY^ß)(a) + m > hßia) ; therefore, gyia) + n> hßia).

Case 2: a < y~iß). Then k+l > hßia) because aß(k) > y~{ß) ; therefore,gy{a) + n> hßia).

Finally, let A c E be nonstationary. Let us use {gy : y < 0} to produceg : 8 —> co that weakly bounds {hß : ß e A}. Let Cc0 be a club such thatC n A = 0. For every ß e E let

S+iß) = min(C \ iß + 1)), S-{ß) = sup(C n ß),

gy(ß) = { f



L 1 otherwise.

Let ß e A. As before, let n = k + m + 1, where k is the least such that

aßik) > o~iß) (notice, ß e A implies that S~iß) < ß) and m is such that

Ss\ß) + m> hß . Let us show that g + n> hß. Let ae Enß .

Case 1: ¿"(a) = S~(ß). So, r3+(a) = S+iß) ; therefore, g(a) = gs^a)(a) =

gs\ß)ia). But gs\ß)ia) + m > hßia) ; therefore, g(a) + n > hßia).

Case 2: 3-(a) < S~(ß). In this case o~(ß) > a. But aß(k) > S~(ß);therefore, ap(k) > a and hence aß(k) ¿ aa(k). So, k + 1 > hß(a) ; therefore,

g(a) + n> hßia). D

We remark here that ->E^ is equiconsistent with a Mahlo cardinal [D, HS],

while -*0(co2) is equiconsistent with a weakly compact cardinal [Ma, Toi, Be]

(and therefore Oico2) does not imply E%2 ). So consistency-wise, a result from

Oico2) is better (more difficult to deny) than one from E%2. On the other hand,

E%2 does not imply D(<y2) (PFA is consistent with E%2 but implies -<0(co2)

[Be]).

Theorem 7. Let k be a supercompact cardinal, let P be the Levy collapse of

k to co2 with countable conditions, and let G be a F-generic filter over V.Then in V[G] the following holds: For every regular 8 > co2 and every 8-

family {hß : ß < 8}, if for all A e [8fl, {hß : ß e A} is weakly bounded,then for every stationary set S c 8 there exists a stationary T c S such that

{hß : ß e T} is weakly bounded.

The proof of Theorem 7 is very much like the proof of Theorem 5. The

following lemma is the analog of the Main Lemma; the rest of the proof of the

theorem is left to the reader.

Lemma. Assume that F is an cox-closed partial order and that 8 > co2 is a

regular cardinal. Assume that ff = {hß : ß < 8} is a 8-family and S c 0 isstationary such that for every stationary T c S the family {hß : ß e T} is notweakly bounded. Assume that G is a F-generic filter over V. Then in V[G],

{hß : ß e S} is not weakly bounded.

Proof. We proceed as in our Main Lemma and assume the lemma is false. LetPo G P and g, a P-name, be such that po Ih "g : 8 —> co weakly bounds

{hß-.ßeS}".Let M be an elementary substructure of //(A) for some large enough regular

A, with S, 0, <%*, F, po, g,and lhP all members of M and M n0 = ß eS.Construct Po > Pi > • • ■ > Pn+\ >••• such that pn+\ e M and an< ß with

p„+x U-^gia^ + nKhnian)".

This process must stop at a finite stage since otherwise one could choose q with

q < Pn for every n e co and get that for every n e co there is a„ < ß such

that q lh "g(a„) + n < hßia„Y, which is impossible.So, there is n e co such that for every q <pn and every a < ß , if q e M,

then q ¥ "£(a) + n< hßia)n. As in the Main Lemma, it follows that for every

a<ß,

Pn lh "¿(a) + n> hß(a)n.



Let T = {ß e S : (Va < ß) pñ lh "¿(a) + n> hß(a)"} . By the elementarily ofM, T is stationary.

Now, for every a e 8, let qa <Pn and ^ G co be such that qa lh "g(a) =

wQ". In V, define f : 8 -> co by /(a) = ma + ñ. Finally, continue as in theMain Lemma to get a contradiction by showing that for every ß e T and every

a< ß, hß(a) < f(a). In particular, / weakly bounds {hß : ß e T} . O

By Theorem 6, in the model of Theorem 7, Eff fails for every regular 8 > co2(this is a result of Shelah). As was the case for Theorems 4 and 5, Theorem 7can also be proved for the Mitchell collapse instead of the Levy collapse. Wefinish with a question.

Question 4. Is it consistent (relative to a large cardinal) that every i.w.b. co2-

family extends!

References

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[GT] G. Gruenhage and Tamaño, handwritten notes.

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[Ma] M. Magidor, Reflecting stationary sets, J. Symbolic Logic 47 (1982), 755-771.

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[Tal] F. D. Tall, Weakly collectionwise Hausdorff spaces, Topology Proc. 1 (1976), 295-304.

[Ta2] _, Topological applications of generic huge embeddings, Trans. Amer. Math. Soc. 341

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[Toi] S. Todorcevic, Partitioning pairs of countable ordinals, Acta Math. 159 (1987), 261-294.

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Department of Mathematics, The University of Kansas, Lawrence, Kansas 66045

Current address, T. Laberge: Department of Mathematics, Union College, Schenectady, New

York 12308E-mail address : labergetfigar. union. edu

Current address, A. Landver: IBM Israel—Science & Technology, MAT AM, Haifa 31905, Israel

E-mail address: landverflhaif asc3. vnet. ibm. com


reflection and weakly collectionwise hausdorff spaces

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