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Topological Partition Relations of the Form ω∗→(Y )1

arXiv:math/9305206v1 [math.LO] 15 May 1993Topological Partition Relations of the Form ω∗→(Y )12W. W. Comfort a,b,c, Wesleyan University (USA)Akio Kato, National Defense Academy (Japan)Saharon Shelah d, Hebrew University (Israel)ABSTRACTTheorem.

The topological partition relation ω∗→(Y )12(a) fails for every space Y with |Y | ≥2c;(b) holds for Y discrete if and only if |Y | ≤c;(c) holds for certain non-discrete P-spaces Y ;(d) fails for Y = ω ∪{p} with p ∈ω∗;(e) fails for Y infinite and countably compact.AMS Classification Numbers: Primary 54A25, 54D40, 05A17; Secondary 54A10,54D30, 04A20Key words and phrases: Topological Ramsey theory; topological partitionrelation; Stone-ˇCech remainder; P-space; countably compact space. [Footnotes]a Presented at the Madison Conference by this co-author.b Member, New York Academy of Sciences.c This author gratefully acknowledges support received from the TechnischeHochschule Darmstadt and from the Deutscher Akademischer Austauschdienst(=DAAD) of the Federal Republic of Germany.d This author acknowledges partial support received from the Basic ResearchFund of the Israeli Academy of Sciences, Publ.

440.1

§1. Introduction.For topological space X and Y we write X ≈Y if X and Y are homeomorphic,and we write f : X ≈Y if f is a homeomorphism of X onto Y .

The “topologicalinclusion relation” is denoted by ⊆h; that is, we write Y ⊆h X if there is Y ′ ⊆Y suchthat Y ≈Y ′.The symbol ω denotes both the least infinite cardinal and the countably infinitediscrete space; the Stone-ˇCech remainder β(ω)\ω is denoted ω∗.For a space X we denote by wX and dX the weight and density character of X,respectively. Following [7], for A ⊆ω we write A∗= (clβ(ω)A)\ω.For proofs of the following statements, and for other basic information ontopological and combinatorial properties of the space ω∗, see [7], [3], [12].1.1.

Theorem. (a) {clβ(ω)A : A ⊆ω} is a basis for the open sets of β(ω); thusw(β(ω)) = c.(b) There is an (almost disjoint) family A of subsets of ω such that |A| = c and{A∗: A ∈A} is pairwise disjoint.

(c) ω∗contains a family of 2c-many pairwise disjoint copies of β(ω). (d) Every infinite, closed subspace Y of ω∗contains a copy of β(ω), so|Y | = |β(ω)| = 2c.

✷For cardinals κ and λ and topological spaces X and Y , the symbol X →(Y )κλmeans that if the set [X]κ of all κ-membered subsets of X is written in the form[X]κ = ∪i<λPi, then there are i < λ and Y ′ ⊆X such that Y ≈Y ′ and [Y ′]κ ⊆Pi. Ourpresent primary interest is in topological arrow relations of the form X →(Y )12 (withX = ω∗).

For spaces X and Y , the relation X →(Y )12 reduces to this: if X = P0 ∪P1,then either Y ⊆h P0 or Y ⊆h P1.The relation X →(Y0, Y1)1 indicates that if X = P0 ∪P1, then either Y0 ⊆h P0 orY1 ⊆h P1.It is obvious that if X and Y are spaces such that Y ⊆h X fails, then X →(Y )122

fails.By way of introduction it is enough here to observe that the classical theorem ofF. Bernstein, according to which there is a subset S of the real line R such that neitherS nor its complement R\S contains an uncountable closed set, is captured by theassertion that the relation R →({0, 1}ω)12 fails; in the positive direction, it is easy to seethat the relation Q →(Q)12 holds for Q the space of rationals.For a report on the present-day “state of the art” concerning topologicalpartition relations, and for references to the literature and open questions, the readermay consult [14], [15], [16].This paper is organized as follows.

§2 shows that ω∗→(Y )12 fails for everyinfinite compact space Y . §3 characterizes those discrete spaces Y for which ω∗→(Y )12,and §4 shows that ω∗→(Y )12 holds for certain non-discrete spaces Y .

§5 shows thatω∗→(Y )12 fails for spaces of the form Y = ω ∪{p} with p ∈ω∗, hence fails for everyinfinite countably compact space Y . The results of §§2–5 prompt several questions, andthese are given in §6.We are grateful to Jan van Mill, K. P. S. Bhaskara Rao, and W. A. R. Weiss forhelpful conversations.We announced some of our results in the abstract [2].

See also [1] for relatedresults.§2. ω∗̸→(Y )12 for |Y | ≥2c.2.1.

Lemma. If Y ⊆h ω∗then |{A ⊆ω∗: A ≈Y }| = 2c.Proof.

The inequality ≥is immediate from Theorem 1.1(c). For ≤, it is enoughto fix (a copy of) Y ⊂ω∗and to notice that since dY ≤wY ≤w(ω∗) = c (by Theorem1.1(a)), the number of continuous functions from Y into ω∗does not exceed|(ω∗)dY | ≤(2c)c = 2c.

✷2.2. Theorem.

If Y is a space such that |Y | ≥2c, then ω∗̸→(Y )12.Proof. We assume Y ⊆h ω∗(in particular we assume |Y | = |ω∗| = 2c) since3

otherwise ω∗̸→(Y )12 is obvious. Following Lemma 2.1 let {Aξ : ξ < 2c} enumerate{A ⊆ω∗: A ≈Y }, choose distinct p0, q0 ∈A0 and recursively, if ξ < 2c and pη, qη havebeen chosen for all η < ξ choose distinctpξ, qξ ∈Aξ\({pη : η < ξ} ∪{qη : η < ξ}).It is then clear, writingP0 = {pξ : ξ < 2c} and P1 = ω∗\P0,that the relations Y ⊆h P0 and Y ⊆h P1 both fail.

✷The following statement is an immediate consequence of Theorems 2.2 and 1.1(d).2.3. Corollary.

The relation ω∗→(Y )12 fails for every infinite compact spaceY . ✷By less elementary methods we strengthen Corollary 2.3 in Theorem 5.14 below.§3.

Concerning the Relation ω∗→(Y )12 for Y Discrete.The very simple result of this section, included in the interest of completeness,shows for discrete spaces Y that ω∗→(Y )12 if and only if Y ⊆h ω∗.3.1. Theorem.

For a discrete space Y , the following conditions are equivalent. (a) |Y | ≤c;(b) ω∗→(Y )1c;(c) ω∗→(Y )12;(d) Y ⊆h ω∗.Proof.

(a) ⇒(b). [Here we profit from a suggestion offered by the referee.] Givenω∗= ∪i

That (b) ⇒(c) and (c) ⇒(d) and clear. (d) ⇒(a).

Theorem 1.1(a) gives |Y | = wY ≤w(β(ω)) = c.§4. ω∗→(Y )12 for Certain Non-Discrete Y .For an infinite cardinal κ we denote by Pκ the ordinal space κ + 1 = κ ∪{κ}topologized to be “discrete below κ” and with a neighborhood base at κ the same as inthe usual interval topology.

That is, a subset U of κ + 1 is open in Pκ if and only ifeither U ⊆κ or some ξ < κ satisfies (ξ, κ] ⊆U.4.1. Theorem.

For cardinals κ ≥ω and m0, m1 < ω, the space Pκ satisfiesP m0+m1κ→(P m0κ , P m1κ )1.Proof. Let P I = X0 ∪X1 and |I| = m0 + m1 and suppose without loss ofgenerality that the point c = ⟨ci⟩i∈I with ci = κ (all i ∈I) satisfies c ∈X0.

LetI = I0 ∪I1 with |I0| = m0, |I1| = m1, and set D = Pκ\{κ}, and for x ∈DI0 defineS(x) = {x} × {y ∈P I1κ : max{xi : i ∈I0} < min{yi : i ∈I1}}.If some x ∈DI0 satisfies S(x) ⊆X1 we have P m1κ≈S(x) ⊆X1 and the proof iscomplete. Otherwise for each x ∈DI0 there is p(x) ∈S(x) ∩X0 and thenP m0κ≈{p(x) : x ∈DI0} ∪{c} ⊆X0,as required.

✷4.2. Corollary.

Every infinite cardinal κ satisfies Pκ × Pκ →(Pκ)12. ✷We say as usual that a topological space X = ⟨X, T ⟩is a P-space if each U ⊆Twith |U| ≤ω satisfies ∩U ∈T , Since (clearly) Pκ is a non-discrete P-space if and only ifcf(κ) > ω, the following theorem shows the existence of a nondiscrete Y such thatX →(Y )12.4.3.

Theorem. Let ω1 ≤κ ≤c satisfy cf(κ) > ω.

Then ω∗→(Pκ)12.Proof. It is a theorem of E. K. van Douwen that every P-space X such thatwX ≤c satisfies X ⊆h ω∗.

(For a proof of this result see [4] or [12].) Thus for κ ashypothesized we have Pκ × Pκ ⊆h ω∗, so the relation ω∗→(Pκ)12 is immediate fromCorollary 4.2.

✷5

4.4. Remarks.

(a) The following simple result, suggested by the proof ofTheorem 4.2, is peripheral to the principal thrust of our paper. Here as usual for a spaceX = ⟨X, T ⟩we denote by PX = ⟨PX, PT ⟩the set X with the smallest topology PTsuch that PT ⊇T and PX is a P-space; thus {∩U : U ⊆T , |U| ≤ω} is a base for PT .Theorem.

For a P-space Y , the following conditions are equivalent. (i) ω∗→(Y )12;(ii) {0, 1}c →(Y )12;(iii) P(ω∗) →(Y )12;(iv) P({0, 1}c) →(Y )12.Proof.

The implications (iii) ⇒(iv) ⇒(i) ⇒(ii) follow respectively from theinclusions P(ω∗) ⊆h P({0, 1}c) ⊆h ω∗⊆h {0, 1}c. (Of these three inclusions the thirdfollows from Theorem 1.1, the first from the third, and the second from van Douwen’stheorem cited above.) That (ii) ⇒(iii) follows from P({0, 1}c) ⊆h ω∗(whenceP({0, 1}c) ⊆h P(ω∗)) and the case A = {0, 1}c, B = Y = PY of this generalobservation: if A →(B)12 then PA →(PB)12.

✷(b) We note in passing the following result, from which (with 4.1) it follows thatfor κ ≥ω the space Pκ satisfies P 2nκ→(Pκ)1n+1.Theorem. Let S be a space such that Sm0+m1 →(Sm0, Sm1)1 for m0, m1 < ω.Then S2n →(S)1n+1 for n < ω.(*)Proof.

Statement (*) is trivial when n = 0, and is given by the case m0 = m1 = 1of the hypothesis when n = 1.Now suppose (*) holds for n = k, and let S2k+1 = ∩k+1i=0 Xi. With Y0 = X0 andY1 = ∪k+1i=1 Xi, it follows from S2k+2k →(S2k, S2k) that there is T ⊆S2k+1 such thatT ≈S2k and either T ⊆Y0 or T ⊆Y1.

In the first case we have S ⊆h T ⊆X0, and in thesecond case from T ⊆∪k+1i=1 Xi and (*) at k there exists i such that 1 ≤i ≤k + 1 andS ⊆h Xi, as required. ✷(c) The method of proof of 4.1 and 4.2 applies to many spaces other than those6

of the form Pκ. The reader may easily verify, for example, denoting by Cκ the one-pointcompactification of the discrete space κ, that Cκ × Cκ →(Cκ)12, and hence{0, 1}κ →(Cκ)12, for all κ ≥ω.

For a proof due to S. Todorˇcevi´c of a much strongertopological partition relation, namely {0, 1}κ →(Cκ)1cf(κ), see Weiss [15].§5. ω∗̸→(Y )12 for Y Infinite and Countably Compact.To prove this result, we show first that the relation ω∗→(ω ∪{p})12 fails forevery p ∈ω∗.

While this can be proved directly by combinatorial arguments, we find itconvenient (given p ∈ω∗) to introduce and use as a tool a new topology T (p) on ω∗.Given f : ω →ω∗, we denote by f : β(ω) →ω∗the Stone extension of f. ForX ⊆ω∗we setXp = X ∪{f(p) : f : ω ≈f[ω] ⊆X};that is, Xp is X together with its “p-limits through discrete countable sets.”5.1. Lemma.

There is a topology T (p) for ω∗such that each X ⊆ω∗satisfies: Xis T (p)-closed if and only if X = Xp.Proof. It is enough to show(a) ∅= ∅p;(b) ω∗= (ω∗)p;(c) X0 ∪X1 = (X0 ∪X1)p if Xi = Xpi (i = 0, 1) and(d) ∩i∈IXi = (∩i∈IXi)p if each Xi satisfies Xi = Xpi .Now (a) and (b) are obvious, as are the inclusions ⊆of (c) and (d).

(c) (⊇) If f : ω ≈f[ω] ⊆X0 ∪X1 satisfies f(p) = x ∈(X0 ∪X1)p then withAi = {n < ω : f(n) ∈Xi} we have A0 ∪A1 ∈p and hence Ai ∈p for suitable i ∈{0, 1};changing the values of f on ω\Ai if necessary (to ensure f[ω] ⊆Ai), we conclude thatx = f(p) ∈Xpi = Xi ⊆X0 ∪X1. (d) (⊇).

If x = f(p) with f : ω ≈f[ω] ⊆∩iXi then x ∈∩i(Xpi ) = ∩iXi. ✷5.2.

Remarks. (a) In the terminology of Lemma 5.1, the topology T (p) is definedby the relation7

T (p) = {ω∗\X : X ⊆ω∗, X is T (p)-closed}. (b) For notational convenience we denote by I(p) the set of T (p)-isolated pointsof ω∗, and we write A(p) = ω∗\I(p).

Clearly x ∈I(p) if and only if x is not a “discretelimit” of points in ω∗\{x}, that is, if and only if every f : ω ≈f[ω] ⊆ω∗\{x} satisfiesf(p) ̸= x. The fact that I(p) ̸= ∅has been known for many years.

Indeed, Kunen [10]has shown that there exist 2c-many points x ∈ω∗such that x /∈clβ(ω)A wheneverA ⊆ω∗\{x} and |A| ≤ω. (These are the so-called weak-P-points of ω∗.

)As a mnemonic device one may think of A(p) and I(p) as the sets of p-accessibleand p-inaccessible points, respectively. (c) For X ⊆ω∗the set Xp may fail to be closed.

Indeed, the T (p)-closure ofX ⊆ω∗is determined by the following iterative procedure (cf. also [1]).5.3.

Lemma. Let X ⊆ω∗.

For ξ ≤ω+ define Xξ by :X0 = X;Xξ = ∪η<ξ Xη if ξ is a limit ordinal;Xξ+1 = Xpξ .Then Xω+ = T (p) −cl X. ✷The following fact, noted in [8], [5], [6], is crucial to many studies of ω∗(see also[3](16.13) for a proof).

One may capture the thrust of this lemma by paraphrasing thepicturesque terminology of Frol´ik [6]: “No type produces itself.”5.4. Lemma.

No homeomorphism from β(ω) into ω∗has a fixed point. ✷5.5.

Lemma. Let A and B be countable, discrete subsets of ω∗, with A ⊆B∗.Then Ap ∩Bp = ∅.Proof.

If x ∈Ap ∩Bp we may suppose without loss of generality that there aref : ω ≈A and g : ω ≈B such that x = f(p) = g(p). The functionh = f ◦g−1 : B ≈A ⊆B∗satisfiesf ◦g−1 = h : β(B) ≈β(A) ⊆B∗and h(x) = x ∈B∗, contrary to Lemma 5.4.

✷8

5.6. Corollary.

Let A and B be countably infinite, discrete subsets of ω∗suchthat A ∩B = ∅. Then Ap ∩Bp = ∅.Proof.

Let x ∈Ap ∩Bp and let f : ω →f[ω] ⊆A and g : ω →g[ω] ⊆B satisfyx = f(p) = g(p). Leaving f and g unchanged on suitably chosen elements of p, butmaking modifications elsewhere if necessary, we assume without loss of generality thateither f[ω] ⊆(g[ω])∗or g[ω] ⊆(f[ω])∗or f[ω] ∩(g[ω])∗= (f[ω])∗∩g[ω] = ∅.

By Lemma5.5 the first of these possibilities, and by symmetry the second, cannot occur. Weconclude that f[ω] ∪g[ω] is a countable, discrete subset of ω∗such that f[ω] ∩g[ω] = ∅;it follows that (f[ω])∗∩(g[ω])∗= ∅, since every countable (discrete) subset of ω∗isC∗-embedded (cf.

[7](14.27, 14N.5), [3](16.15). This contradicts the relationx ∈(f[ω])∗∩(g[ω])∗.

✷5.7. Corollary.

If ω∗⊇X ∈T (p), then Xp ∈T (p).Proof. If ω∗\Xp is not T (p)-closed then there is f : ω ≈f[ω] = A ⊆ω∗\Xp suchthat x = f(p) ∈Xp.

Since X ∈T (p) we have x ∈Xp\X so there isg : ω ≈g[ω] = B ⊆X such that x = g(p). From A ∩B = ∅and 5.6 now followsx ∈Ap ∩Bp = ∅, a contradiction.

✷5.8. Corollary.

If ω∗⊇X ∈T (p) then T (p) −cl X ∈T (p).Proof. This is immediate from 5.3 and 5.7.

✷Our goal is to 2-color the points of ω∗in such a way that every copy of ω ∪{p}receives two colors. First we consider how to extend a given coloring function.5.9.

Lemma. Let ω∗⊇X ∈T (p) and let c : X →2 = {0, 1} be a function withno monochromatic copy of ω ∪{p} (that is, if X ⊇Y ≈ω ∪{p} then c−1({i}) ∩Y ̸= ∅for i ∈{0, 1}).

Then c extends to ˜c : Xp →2 with no monochromatic copy of ω ∪{p}.Proof. Set Xi = c−1({i}) for i ∈2 = {0, 1}, so that Xp = Xp0 ∪Xp1 by 5.1(c) and(Xp0\X) ∩(Xp1\X) = ∅by 5.6.

Since {X , Xp0\X , Xp1\X} is a partition of X, the function ˜c : Xp →2, givenby the rule9

˜c(x) = c(x) if x ∈X= 1 if x ∈Xp0\X= 0 if x ∈Xp1\X,in well-defined. To see that ˜c is as required let h : ω ∪{p} ≈A ∪{x} ⊆Xp withh : ω ≈A, h(p) = x. Modifying h (as before) if necessary, we assume without loss ofgenerality that either (i) A ⊆X0 or (ii) A ⊆X0\X (the cases A ⊆X1, A ⊆Xp1\X aretreated symmetrically).

In case (i) we have ˜c ≡0 on A and ˜c(x) = 1 (since either x ∈Xor x ∈Xp0\X); case (ii) cannot arise, since x ∈X violates X ∈T (p) while x ∈Xp\Xviolates Corollary 5.6. ✷Combining Lemmas 5.9 and 5.3 yields this.5.10.

Lemma. Let ω∗⊇X ∈T (p) and let c : X →{0, 1} be a function with nomonochromatic copy of ω ∪{p}.

Then c extends to ˜c : T (p) −cl X →{0, 1} with nomonochromatic copy of ω ∪{p}. ✷The preceding lemma indicates how to extend a coloring function from X ∈T (p)over T (p) −cl X, but it remains to initiate the coloring procedure.

For this purpose it isconvenient to consider a particular base S(p) for the topology T (p). We call theelements of S(p) the p-satellite sets.5.11.

Definition. Let x ∈ω∗.

A set S = S(x) is a p-satellite set based at x ifthere are a tree T ⊆ω<ω = ∪n<ω ωn (ordered by containment) and for s ∈T a pointxs ∈S and Us ⊆ω∗such that(i) Us is open-and-closed in the usual topology of ω∗;(ii) x = x⟨⟩with ⟨⟩the empty sequence;(iii) U⟨⟩= ω∗;(iv) if xs ∈S(x) and xs ∈A(p) then: {xsˆn : n < ω} enumerates the range of afunction f such that f : ω ≈f[ω] ⊆ω∗with f(p) = xs, and {Usˆn : n < ω} is a pairwisedisjoint family such that xsˆn ∈Usˆn ⊆Us;(v) if xs ∈S(x) and xs ∈I(p) then s is a maximal node in T (and xsˆn, Usˆn are10

defined for no n < ω).5.12. Remark.

It is not difficult to see that for every x ∈X ∈T (p) there isS = S(x) ∈S(p) such that x ∈S ⊆X. (If x ∈I(p) one takes S = {x}; if xs ∈S ∩Xhas been defined one uses (iv) and X ∈T (p) to choose xsˆn ∈S ∩X if xs ∈A(p).) Thateach of the sets S(x) is T (p)-open is immediate from Corollary 5.6 above.

It follows thatS(p) is indeed a base for T (p).5.13. Theorem.

Every p ∈ω∗satisfies ω∗̸→(ω ∪{p})12.Proof. Let {S(x(i)) : i ∈I} be a maximal pairwise disjoint subfamily of S(p).For each i ∈I define ci : S(x(i)) →2 byci(x(i)s) = 0 if length of s is even= 1 if length of s is odd.It is clear from Corollary 5.6 that not only each function ci on S(x(i)), but also thefunctionc = ∪i∈I ci : ∪i∈I S(x(i)) →2,is monochromatic on no copy of ω ∪{p}.

Since ∪i∈I S(x(i)) is T (p)-open andT (p)-dense in ω∗, the desired result follows from Lemma 5.10. ✷5.14.

Theorem. The relation ω∗→(Y )12 fails for every infinite, countablycompact space Y .Proof.

Given infinite Y ⊆ω∗there is f : ω ≈f[ω] ⊆Y , and if Y is countablycompact there is p ∈ω∗such that f(p) ∈Y . Since f[ω] is C∗-embedded in ω∗we haveω ∪{p} ≈f[ω] ∪{f(p)} ⊆Y ,so ω∗̸→(Y )12 follows from ω∗̸→(ω ∪{p})12.

✷5.15. Remarks.

(a) We cite three facts which (taken together) show that theindex set I used in the proof of Theorem 5.13 satisfies |I| = 2c: (i) The set W ofweak-P-points of ω∗introduced by Kunen [10] satisfies |W| = 2c; (ii) each S(x) ∈S(p)satisfies |S(x)| ≤ω; (iii) W ⊆I(p), so W ⊆∪i∈I S(x(i)). (b) With no attempt at a complete topological classification, we note five11

elementary properties enjoyed by each of our topologies T (p) on ω∗. (i) T (p) refines the usual topology of ω∗, so T (p) is a Hausdorfftopology.

(ii) T (p) has 2c-many isolated points. (Indeed, we have noted already thatthe set W of weak-P-points satisfies |W| = 2c and W ⊆I(p).

)(iii) Since S(p) is a base for T (p) and each S(x) ∈S(p) satisfies |S(x)| ≤ω,the topology T (p) is locally countable. (iv) From Theorem 1.1(b) it is easy to see that if S(x) ∈S(p) and |S(x)| = ω,then |T (p) −cl S(x)| = c. Thus T (p) is not a regular topology for ω∗.

(v) According to Corollary 5.8, the T (p)-closure of each T (p)-open subset ofω∗is itself T (p)-open. Such a topology is said to be extremally disconnected.

(c) In our development of T (p) and its properties we did not introduce explicitlythe Rudin-Frol´ik pre-order ⊑on ω∗(see [5], [6], or [13], or [3] for an expositorytreatment) since doing so does not appear to simplify the arguments. We note however(as in [1]) that the relation ⊑lies close to our work: For x, p ∈ω∗one has p ❁x if andonly if some f : ω ≈f[ω] ⊆ω∗satisfies f(p) = x.

✷§6. Questions.Perhaps this paper is best viewed as establishing some boundary conditionswhich may help lead to a solution of the following ambitious general problem.6.1.

Problem. Characterize those spaces Y such that ω∗→(Y )12.

✷There are P-spaces Y such that |Y | = 2c and Y ⊆h ω∗. (For example, accordingto van Douwen’s theorem cited above, one may take Y = P(ω∗).) According to Theorem2.2, the relation ω∗→(Y )12 fails for each such Y .

This situation suggests the followingquestion.6.2. Question.

Does ω∗→(Y )12 for every P-space Y such that Y ⊆h ω∗and|Y | < 2c? What if |Y | = c?

✷We have no example of a non-P-space Y such that ω∗→(Y )12, so we arecompelled to ask:12

6.3. Question.

If Y is a space such that ω∗→(Y )12, must Y be a P-space? ✷For |Y | = ω, Question 6.3 takes this simple form:6.4.

Question. If Y is a countable space such that ω∗→(Y )12, must Y bediscrete?

✷6.5. Remark.

In connection with Question 6.4 it should be noted that thereexists a countable, dense-in-itself subset C of ω∗such that every x ∈C satisfies(*) x /∈clβ(ω) D whenever D is discrete and D ⊆C\{x}(equivalently: ω ∪{p} ⊆h C fails for every p ∈ω∗). To find such C we follow theconstruction of van Mill [11](3.3, pp.

53-54). Let E be the absolute (i.e., the Gleasoncover) of the Cantor set {0, 1}ω, let π : E →{0, 1}ω be perfect and irreducible, andembed E into ω∗as a c-OK set; then every countable F ⊆ω∗\E satisfiesE ∩clβ(ω)F = ∅.

Now by the method of [11](3.3) for t ∈{0, 1}ω choose xt ∈π−1({t})such that every discrete D ⊆E\{xt} satisfies xt /∈clβ(ω)D, and take C = {xt : t ∈C0}with C0 a countable, dense subset of {0, 1}ω. Since π is irreducible the set C is dense inE and is dense-in-itself, and it is easy to see that condition (*) is satisfied.Of course no element of C is a P-point of ω∗.

The existence in ZFC ofnon-P-points x ∈ω∗such that x /∈clβ(ω)D whenever D is a countable, discrete,subspace of ω∗\{x} is given explicitly by van Mill [11]; see also Kunen [9] for aconstruction in ZFC + CH (or, in ZFC + MA) of a set C as above.For the set C constructed above the relation ω ∪{p} ⊆h C fails for every p ∈ω∗,so the following question, closely related to Question 6.4, is apparently not answered bythe methods of this paper.6.6. Question.

Let C be a countable, dense-in-itself subset of ω∗such thatω ∪{p} ⊆h C fails for every p ∈ω∗. Is the relation ω∗→(C)12 valid?

✷13

List of References[1] W. W. Comfort and Akio Kato. Non-homeomorphic disjoint spaces whoseunion is ω∗.

Rocky Mountain J. Math.

To appear. [2] W. W. Comfort and Akio Kato.

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