The Stone-Cech compactifications of ω ∗ ∖{x} and S κ ∖{x}
aa r X i v : . [ m a t h . GN ] M a y COMPACTIFICATIONS OF ω ∗ \ { x } AND S κ \ { x } MAX F. PITZ AND ROLF SUABEDISSEN
Abstract.
The title refers to the Stone-Čech remainder of the integers ω ∗ and the Stone space S κ of the κ -saturated Boolean algebra of cardinality κ .The latter space is characterised topologically as the unique κ -Parovičenkospace of weight κ . It exists if and only if the consistent and independentequality κ = κ <κ holds. The spaces S κ are generalisations of the space ω ∗ :under the Continuum Hypothesis, S ω is homeomorphic to ω ∗ .In this paper we investigate compactifications of spaces S κ \{ x } , building onand extending corresponding results obtained by Fine & Gillman and Comfort& Negrepontis for the space ω ∗ .We show that for every point x of S κ , the Stone-Čech remainder of S κ \{ x } is a κ + -Parovičenko space of cardinality κ which admits a family of κ disjoint clopen sets. As a corollary we get that it is consistent with CH thatthe Stone-Čech remainders of ω ∗ \ { x } are all homeomorphic. Introduction
The purpose of this paper is to study compactifications of subspaces of the Stone-Čech remainder of the integers ω ∗ = βω \ ω , namely subspaces of the form ω ∗ \ { x } .The questions discussed in this paper are motivated by two classical theorems aboutextensions of real-valued continuous functions on ω ∗ \ { x } . Theorem 1.1 (Fine and Gillman, 1960 [8, 10]) . Assuming the Continuum Hypo-thesis, for every point x ∈ ω ∗ there are continuous bounded real-valued functionson ω ∗ \ { x } that cannot be continuously extended to ω ∗ . Theorem 1.2 (van Douwen, Kunen and van Mill, 1989 [3]) . It is consistent with thecontinuum being ℵ that for every point x ∈ ω ∗ all continuous real-valued functionson ω ∗ \ { x } can be continuously extended to ω ∗ . Phrased in the language of compactifications, Theorem 1.2 says that the Stone-Čech compactification of ω ∗ \ { x } consistently coincides with ω ∗ . Theorem 1.1,however, guarantees that under the Continuum Hypothesis (CH), the Stone-Čechcompactification certainly adds more than a single point to ω ∗ \ { x } .This last observation gives rise to a variety of new, interesting questions. Whatare the precise mechanisms that increase the size of the Stone-Čech compactificationof ω ∗ \{ x } under CH? Can one determine its size or is this independent of ZFC+CH?Does the Stone-Čech remainder of ω ∗ \ { x } reflect structural properties of ω ∗ or ω ∗ \ { x } and in particular, does this depend on which point x we remove? Mathematics Subject Classification.
Primary 54D40; Secondary 54D35, 54G05, 06E15.
Key words and phrases.
Stone-Čech compactification, Parovičenko, κ -Parovičenko, κ -saturated Boolean algebra, monotone F -space.The first author acklowledges support of the German Academic Exchange Service (DAAD). What makes these problems particularly interesting is a surprising link betweencompactifications of ω ∗ \ { x } and another research area in the vicinity of ω ∗ : thetheory of κ -Parovičenko spaces, developed by Negrepontis [15], Comfort and Ne-grepontis [2] and Dow [5].A rigorous introduction to κ -Parovičenko spaces will follow in the next section.Intuitively, however, κ -Parovičenko spaces generalise crucial characteristics of ω ∗ tospaces of (potentially larger) weight κ . Recall that under CH, or equivalently under ω = ω <ω , the space ω ∗ is topologically characterised as the unique Parovičenkospace of weight c . Here, the Parovičenko properties entail a precise description ofthe behaviour of countable unions and intersections of clopen sets in ω ∗ : disjointpairs of such countable unions have disjoint closures, and every such non-empty in-tersection has non-empty interior. The κ -Parovičenko properties essentially consistof corresponding requirements for all λ -unions and λ -intersections of clopen sets forall λ < κ . And as for ω ∗ under CH, for every cardinal κ with the property κ = κ <κ there is a unique κ -Parovičenko space of weight κ , denoted by S κ . In particular, S ω coincides with ω ∗ under CH.The main result of this paper is that under CH, the Stone-Čech remainder of ω ∗ \ { x } is an ω -Parovičenko space of cardinality ω and contains a family of ω disjoint open sets, regardless of the choice of x . In general, under κ = κ <κ , theStone-Čech remainder of S κ \ { x } is a κ + -Parovičenko space of cardinality κ andcontains a family of κ disjoint clopen sets. Again, this holds for every point x in S κ .As a consequence, assuming c = ω , the Stone-Čech remainder of any ω ∗ \{ x } ishomeomorphic to the unique ω -Parovičenko space of weight ω , independently ofwhich point x gets removed. This is surprising, considering that subspaces ω ∗ \ { x } and ω ∗ \ { y } are typically very different.Our main result also improves a theorem by A. Dow, who showed in 1985 thatthe remainder of S κ \ { p } is a κ + -Parovičenko space provided that p has a nestedneighbourhood base in S κ , [5]. Consequently, we now have a more comprehensiveanswer to a question by S. Negrepontis [15, p. 522]: under κ + = 2 κ , the space S κ + can be represented as the Stone-Čech remainder of any S κ \ { x } , regardless ofwhether x has a nested neighbourhood base or not.This paper is organised as follows. In Section 2 we recall the relevant backgroundfor the investigation of ω ∗ and S κ . Section 3 provides a short account of Fine andGillman’s result that under CH, no ω ∗ \ { x } is C ∗ -embedded in ω ∗ , and concludeswith a characterisation of the clopen subsets of ω ∗ \ { x } . Section 4 generalisesthese results to spaces S κ for any κ with κ = κ <κ . In Section 5 we show that theStone-Čech compactification of S κ \ { x } is a κ -Parovičenko space of weight κ .The next two sections are concerned with the remainder of S κ \ { x } . First weshow in Section 6 that all remainders are κ + -Parovičenko spaces and conclude thatunder additional set theoretic assumptions they are all homeomorphic. In Section7 we answer some questions about cardinal invariants of these spaces and provethat the Stone-Čech remainder of S κ \ { x } has cardinality κ . Our proof builds onthe observation that under CH, the space ω ∗ is a monotone F-space , i.e. that it ispossible to assign a separating clopen partition to pairs of disjoint open F σ -sets, inthe spirit of a monotone normality operator. We also give an explicit topologicalconstruction of a family of κ disjoint open sets in the Stone-Čech remainder of S κ \ { x } . OMPACTIFICATIONS OF ω ∗ \ { x } AND S κ \ { x } Each of Section 5 to 7 builds only on Section 4 and can be read independently.Section 8 concludes the paper with some open questions.The authors would like to thank Alan Dow for his help with the proof of Theorem6.3 at the Spring Topology and Dynamics Conference 2014 in Richmond, Virginia.2.
Background
We recall properties of ω ∗ and S κ which we use freely in later sections. All thisand more can be found in [2, 11, 14]. The reader should note that our notion ofa ( κ -)Parovičenko space differs from the commonly adopted definition in the sensethat it does not a priori include any assumptions regarding weight.A subset Y of a space X is C ∗ - embedded if every bounded real-valued contin-uous function on Y can be continuously extended to all of X . For a Tychonoffspace X , we write βX for its Stone-Čech compactification , the unique compactHausdorff space in which X is densely C ∗ -embedded, and we write X ∗ = βX \ X for the remainder of X . A space is zero-dimensional if it has a base of clopen(closed-and-open) sets. A space X is strongly zero-dimensional if its Stone-Čechcompactification βX is zero-dimensional. A Tychonoff space X is called F - space if each cozero-set is C ∗ -embedded in X . We list some facts about F -spaces [11,Ch. 14].(1) X is an F -space if and only if βX is an F -space.(2) A normal space is an F -space if and only if disjoint open F σ -sets havedisjoint closures.(3) Closed subspaces of normal F -spaces are again F -spaces.(4) Infinite closed subspaces of compact F -spaces contain a copy of βω . There-fore, compact F -spaces do not contain convergent sequences.In compact zero-dimensional spaces, cozero-sets are countable unions of clopensets. This motivates the following definition from [2, Ch. 14]. In a zero-dimensionalspace X , the ( X -) type of an open subset U of X is the least cardinal number τ = τ ( U ) such that U can be written as a union of τ -many clopen subsets of X .A zero-dimensional space where open subsets of type less than κ are C ∗ -embeddedis called F κ - space . In zero-dimensional compact spaces the notions of F - and F ω -space coincide.( ′ ) X is a strongly zero-dimensional F κ -space if and only if βX is a stronglyzero-dimensional F κ -space. See Theorem 5.1.( ′ ) In F κ -spaces, disjoint open sets of types less than κ have disjoint closures,and in normal spaces both conditions are equivalent, [2, 6.5].The space ω ∗ is a compact zero-dimensional Hausdorff space without isolatedpoints with the following two extra properties: it is an F -space in which eachnon-empty G δ -set has non-empty interior. Such a space is also called Parovičenkospace .We call a space a G κ -space if every non-empty intersection of less than κ -manyopen sets has non-empty interior. Then ω ∗ is a G ω -space.(5) The following are equivalent for a zero-dimensional space X :(a) X is a G κ -space,(b) For every open subset U of X of type less than κ , no set H with U ( H ⊆ U is open, M. PITZ AND R. SUABEDISSEN (c) For every open subset U of X such that < τ ( U ) < κ , its closure U is not open,(d) No open subset U of X with < τ ( U ) < κ is dense in X .For proofs of the non-trivial implications ( a ) ⇒ ( b ) and ( d ) ⇒ ( a ) see [2, 14.5]and [2, 6.6] respectively.We call a space a κ - Parovičenko space if it is a compact zero-dimensional F κ -and G κ -space without isolated points. This modifies the corresponding definitionsof [4] and [5, 1.2], freeing κ -Parovičenko spaces from additional weight restrictions.Under our definition, the space ω ∗ is an ( ω -)Parovičenko space of weight c .It is not hard to prove that any κ -Parovičenko space has weight at least κ <κ .Thus, the following two characterisation theorems say, loosely speaking, that thesmallest possible ( κ -)Parovičenko spaces, namely the ones of weight κ , are topolog-ically unique, and larger ones are not. Theorem 2.1 (Parovičenko [16], van Douwen & van Mill [4]) . CH is equivalent tothe assertion that every ( ω -)Parovičenko space of weight c is homeomorphic to ω ∗ . Theorem 2.2 (Negrepontis [15], Dow [5]) . The assumption κ = κ <κ is equivalentto the assertion that all κ -Parovičenko spaces of weight κ <κ are homeomorphic. If the condition κ = κ <κ is satisfied then the unique κ -Parovičenko space ofweight κ exists and is denoted by S κ [2, 6.12]. Whenever the space S ω exists,it is homeomorphic to ω ∗ . The existence of uncountable cardinals satisfying theequality κ = κ <κ is independent of ZFC but an assumption like κ + = 2 κ impliesthe equality for κ + . Also note that κ = κ <κ implies that κ is regular.A P κ -point p is a point such that the intersection of less than κ -many neigh-bourhoods of p contains again an open neighbourhood of p . A P ω -point is simplycalled P -point. That P κ -points exist in S κ is well-known [2, 6.17]. A new proof ofthis fact is contained in Corollary 4.6. We list some facts about P κ -points.(6) In S κ , p is a P κ -point if and only if p has a nested neighbourhood base ifand only if p is not contained in the boundary of any open set of type lessthan κ .(7) For every pair of P κ -points in S κ there exists an autohomeomorphism of S κ mapping one P κ -point to the other [2, 6.21].In particular, the subspaces S κ \ { p } are homeomorphic for all P κ -points p . Corre-sponding results hold for P -points in ω ∗ under CH.3. The Butterfly Lemma
We begin with the classic result that under CH, ω ∗ does not occur as the Stone-Čech compactification of any of its dense subspaces.A point x of a Hausdorff space X is called strong butterfly point if its complement X \ { x } can be partitioned into open sets A and B such that A ∩ B = { x } . Thesets A and B are called wings of the butterfly point x . Note that in X \ { x } , thewings A and B are clopen and non-compact.The following lemma by Fine and Gillman [8, 10] states that under CH, everypoint in ω ∗ is a strong butterfly point. We outline the proof, as later on we willuse variations of this approach in Theorem 6.3 and Lemmas 6.7 and 7.3. Lemma 3.1 (Butterfly Lemma, Fine and Gillman) . [CH]. Every point in ω ∗ is astrong butterfly point. OMPACTIFICATIONS OF ω ∗ \ { x } AND S κ \ { x } Proof.
Let x ∈ ω ∗ and fix a neighbourhood base { U α : α < ω } of x consisting ofclopen sets. By transfinite recursion we define families of clopen sets { A α : α < ω } and { B α : α < ω } not containing x such that all ( A α , B β )-pairs are disjoint and X \ U α ⊆ A α ∪ B α and A α ∩ U α = ∅ 6 = B α ∩ U α for all α < ω . Once the construction is completed, we define disjoint open sets A = [ α<ω A α and B = [ α<ω B α . Their union covers of all of ω ∗ \ { x } and both A and B limit onto x .It remains to describe the recursive construction. Let α < ω and assume that A β and B β have been defined for all ordinals β < α . Since countable unions ofclopen sets are open F σ -sets, by the F -space property there exist clopen sets C and D partitioning ω ∗ and containing the disjoint sets S β<α A β and S β<α B β respectively.The set U α \ S β<α ( A β ∪ B β ) is a non-empty G δ -set of the Parovičenko space ω ∗ as it contains x and thus, it has non-empty interior. Hence, inside this set wemay find disjoint non-empty clopen sets C ′ and D ′ not containing x . By defining A α = ( C \ U α ) ∪ C ′ and B α = ( D \ U α ) ∪ D ′ we see that A α and B α are asrequired. (cid:3) We list some consequences of the Butterfly Lemma regarding ω ∗ under CH. Firstmust come the result for which the Butterfly Lemma was originally invented. Corollary 3.2 (Fine and Gillman) . [CH]. For every point x of ω ∗ the subspace ω ∗ \ { x } is not C ∗ -embedded in ω ∗ . (cid:3) The construction of the Butterfly Lemma can be used to build ω many distinctclopen subsets of ω ∗ \ { x } . Indeed, in the proof of Lemma 3.1, for every α < ω wehave the choice of adding C ′ or D ′ to A α . The collection of all clopen subsets of aspace X is denoted by CO ( X ) . Corollary 3.3. [CH].
For all x in ω ∗ we have |CO ( ω ∗ \ { x } ) | = 2 ω . (cid:3) Compact clopen sets of ω ∗ \ { x } are of course homeomorphic to ω ∗ . The nextlemma describes how the non-compact clopen sets look like. Lemma 3.4. [CH].
For every x in ω ∗ , the one-point compactification of a clopennon-compact subset of ω ∗ \ { x } is homeomorphic to ω ∗ .Proof. Let A be a clopen non-compact subset of ω ∗ \ { x } . Taking A ∪ { x } , a closedsubset of ω ∗ , as representative of its one-point compactification, we see that it is azero-dimensional compact F -space of weight c without isolated points.For the G ω -space property, suppose that U ⊆ A ∪ { x } is a non-empty G δ -set.If U has empty intersection with A , then the singleton U = { x } is a G δ -set, andhence has countable character in the compact Hausdorff space A ∪ { x } . It followsthat there is a non-trivial sequence in ω ∗ converging to x , a contradiction. Thus, U intersects the open set A and their intersection is a non-empty G δ -set of ω ∗ withnon-empty interior.An application of Parovičenko’s theorem completes the proof. (cid:3) In absence of CH, the above proof still shows that the one-point compactificationof any clopen non-compact subset of ω ∗ \ { x } is a Parovičenko space of weight c . M. PITZ AND R. SUABEDISSEN
It also follows that ω ∗ contains P -points under CH, as the next lemma shows.Even more, we have another proof of [1, 2.3] that under CH for every point x of ω ∗ there is a clopen copy of ω ∗ contained in ω ∗ such that x is a P -point with respectto that copy. Corollary 3.5. [CH].
For every point x of ω ∗ , at least one of its wings togetherwith x itself is a copy of ω ∗ such that x is a P -point with respect to that copy.Proof. This follows from the last lemma together with the observation that if x wasa non- P -point with respect to both of its wings, it would be in the closure of twodisjoint open F σ -sets, contradicting the F -space property of ω ∗ . (cid:3) Butterflies in S κ \ { x } In this section we generalise results from the previous section about ω ∗ = S ω togeneral S κ , assuming κ = κ <κ throughout. The reader is encouraged to check thatthe Butterfly Lemma and its immediate consequences carry over nicely to S κ . Lemma 4.1 (Butterfly Lemma) . Assume κ = κ <κ . Every point of S κ is a strongbutterfly point. (cid:3) Corollary 4.2.
Assume κ = κ <κ . For every point x of S κ the subspace S κ \ { x } is not C ∗ -embedded in S κ . (cid:3) Corollary 4.3.
Assume κ = κ <κ . For every point x in S κ we have |CO ( S κ \ { x } ) | =2 κ . (cid:3) The remaining part of this section is devoted to the proof of the following gen-eralisation of Lemma 3.4.
Lemma 4.4.
Assume κ = κ <κ and let x any point in S κ . The one-point compacti-fication of a clopen non-compact subset of S κ \ { x } is homeomorphic to S κ . The proof, however, is more delicate than in the case of ω ∗ . The challenge lies inthe fact that Lemma 3.4 used as corner stones two facts about F -spaces which donot carry through to general F κ -spaces: In normal spaces, the F -space property isclosed-hereditary and every infinite closed subset of S ω has the same cardinality as S ω . Both assertions do not hold for F κ -spaces since S κ contains, being an infinitecompact F -space, closed copies of βω .The following lemma is crucial in circumventing these obstacles. Lemma 4.5.
Assume κ = κ <κ and let x any point in S κ . A clopen, non-compactsubset of S κ \ { x } is of S κ -type κ .Proof. Suppose for a contradiction that there exists a clopen, non-compact subset A of S κ \ { x } of S κ -type τ < κ . Find a representation A = [ α<τ A α where all A α are clopen subsets of S κ . We first claim that there is a collection { V α : α < τ } of pairwise disjoint clopen sets of S κ such that V α ⊆ A \ S β<α A β forall α < τ .We proceed by transfinite recursion. Choose a clopen subset V in the non-emptyopen set A \ A . Now consider α < τ and suppose that V β have been defined forall β < α . By minimality of τ , the set U α = S β<α A β ∪ V β is a proper subset of A , OMPACTIFICATIONS OF ω ∗ \ { x } AND S κ \ { x } from which it follows by (5) b of Section 2 that U α is not dense in A . Thus, there isa non-empty clopen set V α in the interior of A \ U α . This completes the recursionand proves the claim.Now, let f and g be disjoint cofinal subsets of τ . We define disjoint open sets V f = [ α ∈ f V α and V g = [ α ∈ g V α of type at most τ and claim that both sets limit onto x , contradicting the F κ -spaceproperty of S κ . Suppose the claim was false. Then V f is a subset of A = S α<τ A α .By compactness, there is a finite set F ⊆ τ such that V f ⊆ S β ∈ F A β . But thereare sets V α with arbitrarily large index contributing to V f , a contradiction. (cid:3) Proof of Lemma 4.4.
Let A be a clopen non-compact subset of S κ \{ x } , and denoteby X the closure of A in S κ , i.e. X = A ∪ { x } ⊆ S κ . Then X is a compact zero-dimensional space of weight κ without isolated points. We check for the remaining κ -Parovičenko properties.To show that X has the F κ -space property, let U and V be disjoint open sets of X of type less than κ . By normality, it suffices to show that U and V have disjointclosures in X . First suppose that x belongs to U ∪ V . Assume x ∈ U , so that x does not belong to the closure of V . The sets U ∩ A and V ∩ A are of A -type lessthan κ . And since A is an F κ -space by [2, 14.1], they have disjoint closures in A ,and therefore in X . Next, suppose that x does not belong to U ∪ V . Then U and V are subsets of A , and consequently of S κ -type less than κ . Thus, U and V havedisjoint closures in S κ , and hence in X . This establishes that X is an F κ -space.To show that X has the G κ -space property, suppose that U = T α<λ U α is anon-empty set for λ < κ where all U α are clopen subsets of X = A ∪ { x } . If U has empty intersection with A , then all X \ U α are clopen subsets of S κ . It followsthat A = S α<λ X \ U α is a clopen non-compact subspace S κ \ { x } of type lessthan κ , contradicting Lemma 4.5. Thus, U intersects A , and their intersection hasnon-empty interior in S κ . (cid:3) Following the proof of Corollary 3.5, we see that for every point x of S κ there isa clopen copy of S κ contained in S κ such that x is a P κ -point with respect to thatcopy. In particular, S κ contains P κ -points. Corollary 4.6.
Assume κ = κ <κ . For every point x of S κ and for every butterflyaround x , one of its wings together with x itself is a copy of S κ such that x is a P κ -point with respect to that copy. (cid:3) The Stone-Čech compactification of S κ \ { x } In this section we show that β ( S κ \ { x } ) is a κ -Parovičenko space of large weight.The result is best possible in the sense that β ( S κ \ { x } ) cannot be a κ + -Parovičenkospace, for points in S κ \ { x } continue to have character κ .We first prove a theorem about F κ -spaces which generalises the well-known corre-sponding theorem for F -spaces (compare ( ) and ( ′ ) in Section 2). It is interestingto note that the F -space property is even hereditary with respect to C ∗ -embeddedsubspaces, but the same is not true for F κ -spaces—see the remarks after Lemma4.4. Theorem 5.1.
A strongly zero-dimensional space is an F κ -space if and only if itsStone-Čech compactification is an F κ -space. M. PITZ AND R. SUABEDISSEN
Proof.
Suppose that X is an F κ -space, and U an open set of βX of type less than κ . To show that U is C ∗ -embedded in βX , fix a continuous [0 , -valued function f defined on U . The set U ∩ X is of type less than κ in X . By assumption, f | U ∩ X can be extended to a continuous [0 , -valued function F on X which again can beextended to a continuous [0 , -valued function βF on βX . Now βF | U = f , as bothfunctions agree on the dense subset U ∩ X .For the converse, assume that βX is an F κ -space. We show by induction on λ that X is an F λ -space for all λ ≤ κ . For λ = ω there is nothing to prove. So let λ ≤ κ be uncountable and assume that X is an F µ -space for all µ < λ . Let U bean open set of X -type τ < λ . We aim to show that U is C ∗ -embedded in X .There are X -clopen sets U α such that U = S α<τ U α . Write V β = S α<β U α and W β = S α<β U α where the closure is taken in βX . Note that all U α are clopensubsets of βX and that V β = W β ∩ X . Let f be a continuous [0 , -valued map on V τ = U . For each β < τ , the set V β is C ∗ -embedded in X by induction hypothesis,and hence, f | V β extends to X and then to βX . Let f β be the restriction of thisextension to W β .Since V β = W β ∩ X is dense in W β for all β < τ , the function f τ = S β<τ f β iswell-defined on W τ . And since every U a is a clopen subset of βX , is is not hardto check that f τ is continuous. Thus, f τ extends from W τ to βX by the F κ -spaceproperty. The restriction of this extension to X is the required extension of f . (cid:3) Theorem 5.2.
Assume κ = κ <κ . For every point x of S κ the space β ( S κ \ { x } ) isa κ -Parovičenko space of weight κ .Proof. Let X = S κ \ { x } . First we verify the κ -Parovičenko properties. Clearly, βX does not contain isolated points. By [2, 14.1], every open subspace of S κ is astrongly zero-dimensional F κ -space and hence so is βX by Theorem 5.1.For the G κ -space property, let U = T α<λ U α be a non-empty intersection ofclopen sets in βX for some λ < κ . To prove that U has non-empty interior itsuffices to show that it intersects X .Assume for a contradiction that λ is minimal such that U has empty intersectionwith X . Consider the sets V β = T α<β U α ∩ X . Then T β<λ V β is empty, whereas,without loss of generality, V β \ V β +1 is non-empty for all β < λ . This last set is anon-empty intersection of less than κ -many open sets in S κ and therefore containsa compact clopen set W β .Let f and g be disjoint cofinal subsets of λ . We define disjoint open sets W f = [ α ∈ f W α and W g = [ α ∈ g W α of type at most λ < κ and claim that in S κ both sets limit onto x , contradictingthe F κ -space property. Suppose the claim was false, e.g. that W f does not limitonto x . Then the closure of W f in S κ , a compact set, would be contained in X = S β<λ X \ V β . But by construction of W f , this open cover of W f does not havea finite subcover, giving the desired contradiction.It remains to calculate the weight of β ( S κ \ { x } ) . By Theorem [2, 2.21] andLemmas [2, 2.23(b) & 2.24], the weight of βX for a strongly zero-dimensional space X is equal to the cardinality of CO ( X ) . Now apply Corollary 4.3. (cid:3) OMPACTIFICATIONS OF ω ∗ \ { x } AND S κ \ { x } It is an interesting question whether it is a ZFC theorem that the Stone-Čechcompactification of ω ∗ \ { x } is a Parovičenko space. A. Dow showed that the as-sertion that all open subspaces of ω ∗ are strongly zero-dimensional F -spaces isequivalent to CH [6]. However, we are only interested in open subspaces of theform ω ∗ \ { x } . And these may be strongly zero-dimensional F -spaces even underthe negation of CH, for example when ω ∗ \ { x } is C ∗ -embedded in ω ∗ .The result about the weight also follows from the fact that the remainder of S κ \ { x } has weight κ , see Theorem 7.5.In the case of κ = ω , a tempting proof of Theorem 5.2 can be obtained byobserving that ω ∗ \ { x } is pseudocompact (as it cannot contain a closed copy of ω ),implying that every G δ -set of β ( ω ∗ \ { x } ) intersects ω ∗ \ { x } . But this approachdoes not seem to generalise to S κ \ { x } without extra effort. At the same time, bothapproaches are somehow intertwined: the proof of Theorem 5.2 shows that S κ \ { x } is α -pseudocompact for all α < κ [12, 2.2]. For more on α -pseudocompactness see[12]. 6. The structure of ( S κ \ { x } ) ∗ The previous section has shown that β ( S κ \{ x } ) is a κ -Parovičenko space of quitelarge weight. Thus, these spaces are too large for the κ -Parovičenko properties toprovide meaningful topological restrictions on the variety of potential spaces of thatsize.For a better understanding we therefore turn to an investigation of the remain-ders of these spaces. The main result of this section is that every space of the form ( S κ \ { x } ) ∗ is a κ + -Parovičenko space of weight κ , regardless of the choice of x .It follows that under κ = κ + , all such remainders are homeomorphic to S κ + . Inparticular, it is consistent with CH that all remainders of spaces of the form ω ∗ \{ x } are homeomorphic.Note that by Theorem 5.2, ( S κ \ { x } ) ∗ is a compact zero-dimensional space. Thenext lemma is the first step for establishing the remaining Parovičenko properties. Lemma 6.1.
Assume κ = κ <κ . For every x ∈ S κ the space ( S κ \ { x } ) ∗ has noisolated points.Proof. Suppose that z ∈ ( S κ \{ x } ) ∗ is isolated. Then there is a clopen non-compactsubset A of S κ \ { x } such that A ∪ { z } is compact and A is C ∗ -embedded in A ∪ { z } .By Lemma 4.4, the set A is homeomorphic to S κ \ { y } for some y ∈ S κ . However,this space does not have a one-point Stone-Čech compactification by Corollary 4.2,a contradiction. (cid:3) Our next observation is that for a P κ -point p , the space S κ \ { p } can be writtenas an increasing union of κ -many compact clopen sets, each homeomorphic to S κ .And remainders of increasing unions of κ -Parovičenko spaces are well understood:the next theorem follows from a result by A. Dow from 1985 [5, 2.2]. Theorem 6.2 (Dow) . Assume κ = κ <κ . For a P κ -point p of S κ the space ( S κ \{ p } ) ∗ is a κ + -Parovičenko space. (cid:3) Our key result is the following strengthening of Theorem 6.2.
Theorem 6.3.
Assume κ = κ <κ . For every point x of S κ the space ( S κ \ { x } ) ∗ isa κ + -Parovičenko space of weight κ . We remark that our proof of Theorem 6.3 makes essential use of Dow’s originaltheorem, and does not handle all cases simultaneously. Before presenting the proof,we discuss some interesting corollaries.
Corollary 6.4.
Assume κ = κ <κ . The Stone-Čech compactifications β ( S κ \ { x } ) and β ( S κ \ { y } ) are homeomorphic if and only if S κ \ { x } and S κ \ { y } are homeo-morphic.Proof. By Theorem 6.3, points in the ground space can be distinguished from pointsin the remainder of β ( S κ \ { x } ) by their character. Thus, any homeomorphismbetween β ( S κ \{ x } ) and β ( S κ \{ y } ) restricts to a homeomorphism between S κ \{ x } and S κ \ { y } . (cid:3) Corollary 6.5.
Assume κ = κ <κ and κ = κ + . For every point x the space ( S κ \ { x } ) ∗ is homeomorphic to S κ + .Proof. Write λ = κ + . The condition κ = κ + implies λ = λ <λ . Thus, for every x in S κ , the space ( S κ \{ x } ) ∗ is a λ -Parovičenko space of weight λ = λ <λ by Theorem6.3, and hence, by Negrepontis’ characterisation, homeomorphic to S λ . (cid:3) Corollary 6.6.
Under the cardinal assumption c = ω , the remainders of ω ∗ \ { x } and ω ∗ \ { y } are homeomorphic for all points x and y of ω ∗ . (cid:3) This is especially interesting when compared to the fact that there are c non-homeomorphic subspaces of the form ω ∗ \ { x } , an observation that follows easilyfrom Frolík’s result [9] that there are c orbits in ω ∗ under its autohomeomorphismgroup, i.e. that ω ∗ is badly non-homogenous.The remaining part of this section is devoted to the proof of Theorem 6.3. Forthis, we first need a lemma about separation of disjoint open sets of small type in S κ \ { x } . Note that a clopen non-compact set of S κ \ { x } is of S κ -type κ by Lemma4.5, but of ( S κ \ { x } ) -type . Lemma 6.7.
Assume κ = κ <κ . For any two disjoint open sets V and W of ( S κ \ { x } ) -type less than κ , there is a clopen set A of S κ \ { x } such that V ⊆ A and W ∩ A = ∅ .Proof. The space S κ \ { x } is an F κ -space by [2, 14.1], so V and W have disjointclosures in S κ \ { x } . If V and W have disjoint closures in S κ , by compactness thereis a clopen set A ⊆ S κ separating V from W . Clearly, the intersection of A with S κ \ { x } is as required.So assume that both V and W limit onto x . We build a butterfly such that itswings separate V and W . Let { U α : α < κ } be a clopen neighbourhood base for x in S κ . By transfinite recursion we define S κ -clopen sets A α and B α for α < κ notcontaining x such that all all ( A β , B γ ) -, ( A β , W ) - and ( V, B γ ) -pairs are disjoint and A β ∪ B β covers S κ \ U β for all β, γ < κ .Once the construction is completed, we define disjoint open sets A = S α<κ A α and B = S α<κ B α . Their union covers of all of S κ \ { x } and V ⊆ A and W ⊆ B as required.It remains to complete the recursive construction. Let α < κ and assume that A β and B β have been defined for all ordinals β < α satisfying the inductive re-quirements. Since V is of ( S κ \ { x } ) -type less than κ , it follows easily that V \ U α OMPACTIFICATIONS OF ω ∗ \ { x } AND S κ \ { x } is of S κ -type less than κ . So the sets ( V ∪ [ β<α A β ) \ U α and ( W ∪ [ β<α B β ) \ U α are disjoint open sets of S κ -type less than κ , and by the F κ -space property thereis a clopen partition ( C, D ) of S κ separating them. We put A α = C \ U α and B α = D \ U α , preserving the inductive assumptions. (cid:3) Proof of Theorem 6.3.
Because of Theorem 6.2, it suffices to prove the theorem fornon- P κ -points x . So let us fix a non- P κ -point x of S κ and an open subset V ⊆ S κ of type less than κ that contains x in its boundary. By the F κ -space property, V is C ∗ -embedded in S κ . In particular, if we write X = S κ \ { x } then the closure of V in X has a one-point Stone-Čech compactification. This means the set V limitsonto precisely one point in the remainder of X . For the remaining parts of thisproof, we denote this unique point in V βX \ X by ⋆ . Claim 1.
For every clopen non-empty set C of X ∗ not containing ⋆ there is aclopen non-compact set D ⊆ X which misses V such that D ∗ = D \ D = C .Moreover, every such D is homeomorphic to S κ \ { p } for a P κ -point p . To see that the claim holds, let C be a clopen subset of X ∗ not containing ⋆ and find a clopen non-compact subset E of X with E ∗ = C . There is a clopenneighbourhood U of x in S κ such that U ∩ ( E ∩ V ) = ∅ : otherwise, the closure of E ∩ V in βX would grow into the remainder. But E ∩ V βX \ X ⊆ E βX \ X ∩ V βX \ X = C ∩ { ⋆ } = ∅ , a contradiction. Hence, for some suitable U , the clopen non-compact set D = E ∩ U of S κ does not intersect V . And since the symmetric difference of D and E iscompact, it follows from [2, 2.6d] that D ∗ = E ∗ = C , as claimed.To see that D is homeomorphic to S κ \ { p } for a P κ -point p , note that D and X \ D form a butterfly around x in S κ such that V witnesses that x is not a P κ -pointof S κ \ D . Hence, D is homeomorphic to S κ \ { p } for a P κ -point p by Corollary 4.6,completing the proof of Claim 1. Claim 2.
Every compact clopen set of X ∗ \ { ⋆ } is a κ + -Parovičenko space. By Claim 1, a compact clopen set of X ∗ \ { ⋆ } is of the form ( S κ \ { p } ) ∗ for a P κ -point p of S κ . Claim 2 now follows from Theorem 6.2. Claim 3.
The point ⋆ is a P κ + -point of X ∗ . To prove Claim 3 we show that whenever { C α : α < κ } is a collection of X ∗ -clopen sets not containing ⋆ , there is a clopen set B ∗ ⊆ X ∗ not containing ⋆ suchthat S α<κ C α ⊆ B ∗ .From Claim 1, we know that for every C α there is a clopen non-compact subset D α ⊆ X such that D α ∩ V = ∅ and D ∗ α = C α . In X , we write F ⊆ ∗ G (read: F is almost contained in G ) if there is a clopen neighbourhood U of x in S κ , such that F ∩ U ⊆ G . Write F = ∗ G if F ⊆ ∗ G and G ⊆ ∗ F . We will use the well-knownfact that C α ⊆ C β in X ∗ if and only if D α ⊆ ∗ D β in X .Similar to the proof of Lemma 6.7, our aim is to build a butterfly in X withwings A and B such that V ⊆ A and D α ⊆ ∗ B for all α < κ . Clearly then, B ∗ isas required. To construct such a butterfly, fix a neighbourhood base { U α : α < κ } of x in S κ consisting of clopen sets. By recursion we will define families of S κ -clopen sets { A α : α < κ } and { B α : α < κ } and a third family { E α : α < κ } of X -clopen setssuch that for all α, β < κ (1) A α ∩ B β = ∅ and A α ∪ B α = S κ \ U α ,(2) A α ∩ E β = ∅ and B α ∩ V = ∅ ,(3) E α ⊆ D α and E α = ∗ D α .Once the construction is completed, it follows from (1) that A = S α<κ A α and B = S α<κ B α partition X into disjoint open sets. Condition (2) guarantees both V ⊆ A and E α ⊆ B and finally, condition (3) gives D α ⊆ ∗ B as desired.It remains to complete the recursive construction. Let α < κ and assume that A β , B β and E β have been defined for all ordinals β < α satisfying the inductiveassumptions. The set S κ \ S β<α A β is an intersection of less than κ -many clopensets in S κ containing x . In particular, it is a non-empty intersection of less than κ -many clopen sets in D α ∪ { x } . But by Claim 1, the point x is a P κ -point withrespect to D α ∪ { x } , and hence there is a D α ∪ { x } -clopen neighbourhood E ′ α of x in this space such that E ′ α ⊆ S κ \ S β<α A β . Now put E α = E ′ α \ { x } and note that E α = ∗ D α .By Lemma 6.7 there exist X -clopen sets C and D partitioning X and containingthe disjoint open sets V ∪ [ β<α A β and [ β<α B β ∪ [ β ≤ α E β respectively. By defining A α = C \ U α and B α = D \ U α it is clear that A α , B α and E α satisfy the inductive assumptions (1) - (3) . The proof of Claim 3 is complete. Claim 4.
The space X ∗ is a κ + -Parovičenko space. For this, note that X ∗ is a zero-dimensional compact space without isolatedpoints by Lemma 6.1. Thus it only remains to check for the F κ + - and the G κ + -space property. So suppose one of these conditions fails. This is witnessed by somepoint x . By Claim 2, x must be ⋆ , but this is a contradiction as ⋆ is a P κ + -pointby Claim 3. This proves Claim 4.To complete the proof, it remains to establish the weight of X ∗ . But every κ + -Parovičenko space has weight at least κ , which can either be shown directly byembedding a binary tree of clopen sets of height κ + 1 , or, in the case of X ∗ , beconcluded from Theorem 7.5. (cid:3) Cardinal invariants of ( S κ \ { x } ) ∗ The results from the previous section have shown that under κ = κ + , all re-mainders of the form ( S κ \ { x } ) ∗ are homeomorphic to S κ + , which settles all furthertopological questions regarding cardinality, weight and cellularity of these spaces.However, we do not yet know what happens in absence of κ = κ + . In this sec-tion we therefore investigate cardinal invariants of ( S κ \ { x } ) ∗ without additionalset-theoretic assumptions beyond its existence, i.e. κ = κ <κ .The next lemma says that for questions such as size, weight and cellularity ofthe remainder of S κ \ { x } , it is enough to focus on the remainder of S κ \ { p } for a P κ -point p . OMPACTIFICATIONS OF ω ∗ \ { x } AND S κ \ { x } Lemma 7.1.
Assume κ = κ <κ . For all x ∈ S κ the space ( S κ \ { x } ) ∗ contains aclopen copy of ( S κ \ { p } ) ∗ for a P κ -point p .Proof. By Corollary 4.6, the space S κ \ { x } contains a clopen copy of S κ \ { p } fora P κ -point p . (cid:3) Every point of a κ -Parovičenko space has character at least κ and hence thewhole space has, by [7, 3.12.11], cardinality at least κ . Thus, Lemma 7.1 andTheorem 6.2 give us the inequality κ + ≤ | ( S κ \ { p } ) ∗ | ≤ | ( S κ \ { x } ) ∗ | ≤ κ . In particular, we see once again that under κ = κ + , the cardinality of ( S κ \{ x } ) ∗ is clear. In the remaining part of this paper we show without additional set-theoretic assumptions that ( S κ \{ p } ) ∗ —and hence ( S κ \{ x } ) ∗ for all x —has maximalcardinality. We also provide a topological construction showing that ( S κ \ { x } ) ∗ has maximal cellularity κ . Although this last result can also be concluded directlyfrom Theorem 6.2, it is interesting to see the topological similarities to the relationbetween MAD families on ω and the cellularity of ω ∗ .Before beginning with our preparations we remark that Comfort and Negrepontisobserved in [1, 2.4] that under CH, the space ( ω ∗ \ { p } ) ∗ has cardinality ω . Theirmethod can also be used to determine the cellularity of ( ω ∗ \ { p } ) ∗ . However, theproof does not generalise to S κ for κ > ω . Indeed, it seems to be a recurringnuisance that elegant proofs about ω ∗ that at their heart invoke the Stone-Čechproperties of ω ∗ = βω \ ω do not carry over to general S κ . In the following wepresent an approach that is solely based on the Parovičenko properties.The next lemma guarantees the existence of a monotone cut operator for allpairs of disjoint open sets of type less than κ in S κ , in the spirit of a separationoperator for monotone normality. We make this definition precise. For an orderedpair h A, B i of disjoint open sets of S κ of type less than κ , we define a cut betweenthem to be a clopen set C h A,B i such that A ⊆ C h A,B i ⊆ S κ \ B. Cuts in S κ exist by the F κ -space property. We write h A, B i ≤ h A ′ , B ′ i if A ⊆ A ′ and B ⊇ B ′ . A cut operator C is called monotone if h A, B i ≤ h A ′ , B ′ i implies C h A,B i ⊆ C h A ′ ,B ′ i . The cut operator is called symmetric if C h A,B i = S κ \ C h B,A i .We need the following strengthening of the concept of a monotone cut operator.Let { U α : α < κ } be a decreasing neighbourhood base of clopen sets of a P κ -point p such that U = S κ . Call two subsets A and B of S κ γ - equivalent if A ∩ U γ = B ∩ U γ for some γ < κ . Also, call A a γ - subset of B , and write A ⊆ γ B , if A ∩ U γ ⊆ B ∩ U γ .We extend this idea to capture “local monotonicity” and write h A, B i ≤ γ h A ′ , B ′ i if A ⊆ γ A ′ and B ⊇ γ B ′ . Note that for γ ≤ δ < κ , if h A, B i ≤ γ h A ′ , B ′ i holds thenso does h A, B i ≤ δ h A ′ , B ′ i .A cut operator C with the property such that for all γ , h A, B i ≤ γ h A ′ , B ′ i implies C h A,B i ⊆ γ C h A ′ ,B ′ i will be called a strong monotone cut operator with respect to { U α } . Every strong monotone cut operator is also monotone. Lemma 7.2.
Assume κ = κ <κ . Let F be the collection of all ordered pairs ofdisjoint open sets of S κ of type less than κ . For every decreasing neighbourhoodbase of clopen sets { U α : α < κ } of a P κ -point in S κ with U = S κ , there exists asymmetric strong monotone cut operator operator C : F → CO ( S κ ) with respect to { U α } . Proof. By κ = κ <κ we may list F = {h A α , B α i : α < κ } , such that permuted pairsare next to each other. Let { U α : α < κ } be a decreasing neighbourhood base ofa P κ -point p consisting of clopen sets such that U = S κ . In addition we define U κ = ∅ , obtaining the technical advantage that for all pairs of sets, one is a γ -subsetof the other for some γ ≤ κ .We define an operator C : F → CO ( S κ ) that satisfies for all ordinals β, δ < κ : ( Cut ) A β ⊆ C h A β ,B β i ⊆ S κ \ B β , ( Sym ) C h A β ,B β i = S κ \ C h B β ,A β i and ( M on ) ∀ γ < κ ( h A δ , B δ i ≤ γ h A β , B β i ⇒ C h A δ ,B δ i ⊆ γ C h A β ,B β i ) . We proceed by transfinite recursion. Let α < κ and suppose we have definedcuts C h A β ,B β i for all β < α satisfying the inductive assumptions for all β, δ < α .Consider h A α , B α i . If α is a successor and h A α , B α i = h B α − , A α − i we de-fine C h A α ,B α i = S κ \ C h A α − ,B α − i . This assignment takes care of ( Sym ) and astraightforward calculation shows that also ( Cut ) and ( M on ) are satisfied.Otherwise, for all β < α we let γ ↓ β and γ ↑ β be the least ordinals such that h A β , B β i ≤ γ ↓ β h A α , B α i and h A α , B α i ≤ γ ↑ β h A β , B β i . This is well-defined, as theserelations are satisfied for at least κ . Let C ↓ α = n U γ ↓ β ∩ C h A β ,B β i : β < α o and C ↑ α = n U γ ↑ β \ C h A β ,B β i : β < α o . The idea is that these sets contain all parts of the previously defined cuts wehave to be aware of in order to make our operator respect ( M on ) . Both sets havecardinality less than κ and consist of clopen sets.We claim that the sets A α ∪ ( S C ↓ α ) and B α ∪ ( S C ↑ α ) are disjoint open sets oftype less than κ . They are clearly open and of type less than κ .We demonstrate only that S C ↓ α and S C ↑ α are disjoint, since the other cases aresimilar. For this we show that for any β, δ < α , the sets U γ ↓ β ∩ C h A β ,B β i ∈ C ↓ α and U γ ↑ δ \ C h A δ ,B δ i ∈ C ↑ α have empty intersection. By construction we have h A β , B β i ≤ γ ↓ β h A α , B α i and h A α , B α i ≤ γ ↑ δ h A δ , B δ i . With γ denoting the larger of γ ↓ β and γ ↑ δ we may apply condition ( M on ) to h A β , B β i ≤ γ h A δ , B δ i and obtain C h A β ,B β i ⊆ γ C h A δ ,B δ i . In particular, the sets U γ ∩ C h A β ,B β i and U γ \ C h A δ ,B δ i have empty intersection, and since U γ = U γ ↓ β ∩ U γ ↑ δ , the result follows.Now, since A α ∪ ( S C ↓ α ) and B α ∪ ( S C ↑ α ) are disjoint open sets of S κ of typeless than κ , there exist clopen sets containing the first set and not intersecting thesecond. Choose one and denote it by C h A α ,B α i .This assignment clearly satisfies ( Cut ) , so it remains to check for ( M on ) . Let β < α and suppose h A β , B β i ≤ γ h A α , B α i for some γ < κ . Since we chose γ ↓ β minimal, we have U γ ⊆ U γ ↓ β . By construction we have U γ ↓ β ∩ C h A β ,B β i ⊆ S C ↓ α ⊆C h A α ,B α i and therefore also U γ ∩ C h A β ,B β i ⊆ C h A α ,B α i . Thus, C h A β ,B β i ⊆ γ C h A α ,B α i .The case h A β , B β i ≥ γ h A α , B α i is similar and the proof is complete. (cid:3) We now consider a variation of the butterfly construction which is tailored to P κ -points. Let { U α : α < κ } be a decreasing neighbourhood base of a P κ -point p of S κ with U = S κ . We work through the “onion rings” D α = U α \ U α +1 and assignthem either to the A - or the B -wing, following certain patterns. When compared to OMPACTIFICATIONS OF ω ∗ \ { x } AND S κ \ { x } the original butterfly construction in Lemma 3.1, this adaptation has the advantagethat one does not need to assign cuts at successor stages.The support of a binary sequence f : κ → is the set f − ( { } ) . Lemma 7.3.
Assume κ = κ <κ and let p be a P κ -point of S κ . There is a family (cid:8) A f : f ∈ κ (cid:9) of clopen subsets of S κ \ { p } such that for all f, g ∈ κ (1) A f is non-empty whenever f has non-empty support,(2) A f is non-compact whenever f has unbounded support,(3) A − f ∩ A f = ∅ ,(4) if f ≤ g (pointwise) then A f ⊆ A g and(5) if f = g eventually then there exists a clopen neighbourhood U of p suchthat A f ∩ U = A g ∩ U .Proof. Let { U α : α < κ } be a decreasing neighbourhood base of p with U = S κ ,and let D α = U α \ U α +1 . Let C denote a fixed strong monotone cut operator fromLemma 7.2 with respect to { U α } . For each sequence f ∈ κ we build a butterflywith wings A f and B f around p . As always, this involves defining S κ -clopen sets A fα and B fα for α < κ such that all ( A fα , B fβ ) -pairs are disjoint and A fα ∪ B fα covers S κ \ U α for all α . The rules for the recursive construction are: for all ordinals α < κ set A fα +1 = ( A fα ∪ D α , if f ( α ) = 1 ,A fα , if f ( α ) = 0 , B fα +1 = ( B fα , if f ( α ) = 1 ,B fα ∪ D α , if f ( α ) = 0 , and if λ < κ is a limit ordinal, put A fλ = C [ β<λ A fβ , [ β<λ B fβ \ U λ and B fλ = ( S κ \ A fλ ) \ U λ . The sets A f = S α<κ A fα and B f = S α<κ B fα are disjoint open and cover all of S κ \ { p } . Thus, they define clopen subsets of S κ \ { p } .We claim the sets A f satisfy assertions (1)-(5). It is clear that (1) is satisfied.Next, if f has unbounded support, then A f limits onto p , i.e. is non-compact.For (3) and (4), one shows by induction that A − fα = B fα and A fα ⊆ A gα whenever f ≤ g , using that the cut operator is symmetric and monotone, respectively.For (5) suppose there exists an ordinal δ < κ such that f ( α ) = g ( α ) for all α ≥ δ .We show by induction that A fα ∩ U δ = A gα ∩ U δ . The claim is trivially true for α < δ .So let α ≥ δ and assume the claim holds for all smaller ordinals. The situation isclear for successors, so assume that α is a limit. By induction hypothesis, thepairs h S β<α A fβ , S β<α B fβ i and h S β<α A gβ , S β<α B gβ i are δ -equivalent and hence itfollows from the properties of the cut operator that A fα ∩ U δ = C ( [ β<α A fβ , [ β<α B fβ ) ∩ ( U δ \ U α )= C ( [ β<α A gβ , [ β<α B gβ ) ∩ ( U δ \ U α ) = A gα ∩ U δ . This completes the induction step and the proof. (cid:3)
We now show how to use a family with properties (1)–(5) of Lemma 7.3 to pushultrafilters and almost disjoint families from κ through to the space ( S κ \ { p } ) ∗ .For a subset U of κ , let U ∈ κ denote its characteristic function. Theorem 7.4.
Assume κ = κ <κ . For all x ∈ S κ the space ( S κ \ { x } ) ∗ has cardi-nality κ .Proof. The upper bound is clear by [7, 3.5.3]. Let p be a P κ -point of S κ . By Lemma7.1 it suffices to show that there are at least κ -many z-ultrafilters on S κ \ { p } .Let (cid:8) A f : f ∈ κ (cid:9) be a family of clopen sets of S κ \ { p } with properties (1), (3)and (4) of Lemma 7.3. For an ultrafilter U on κ , consider the family Φ( U ) = (cid:8) A U : U ∈ U (cid:9) . By (1) and (4), Φ( U ) is a filter base for some clopen filter on S κ \ { p } .Let us see that if U and U ′ are distinct ultrafilters on κ , then Φ( U ) and Φ( U ′ ) can only be extended to distinct z-ultrafilters on S κ \ { p } . Indeed, there is U ⊆ κ such that U ∈ U and κ \ U ∈ U ′ . By (3), A U has empty intersection with A κ \ U ,and therefore Φ( U ) and Φ( U ′ ) cannot be extended to the same z-ultrafilter.As there are κ ultrafilters on κ the result follows. (cid:3) Theorem 7.5.
Assume κ = κ <κ . For all x ∈ S κ the space ( S κ \ { x } ) ∗ contains afamily of κ many disjoint clopen sets.Proof. Let p be a P κ -point of S κ . By Lemma 7.1 it suffices to prove the theoremfor ( S κ \ { p } ) ∗ .Let (cid:8) A f : f ∈ κ (cid:9) be a family of clopen sets of S κ \ { p } with properties (2)–(5)of Lemma 7.3. Recall that κ = κ <κ if and only if κ is regular and κ = 2 <κ [2, 1.27].In particular, from κ = 2 <κ we may conclude that there is an almost disjoint family E of size κ on κ , i.e. a family whose members are subsets of κ of full cardinalitysuch that the intersection of any two elements is of size less than κ [13, II.1.3]. Byproperty (2), the family Φ( E ) = (cid:8) A E : E ∈ E (cid:9) consists of non-compact clopen sets of S κ \ { p } . We claim they have pairwisecompact intersection. By regularity of κ , the intersection of distinct elements E and F of E is bounded by an ordinal δ < κ . The function f : κ → , α ( F ( α ) if α ≤ δ, − E ( α ) if α > δ, satisfies F ≤ f , and f = 1 − E eventually. By property (5), there is a clopenneighbourhood U of p such that A f ∩ U = A − E ∩ U . Then A E ∩ A F ∩ U ⊆ A E ∩ A f ∩ U = A E ∩ A − E ∩ U = ∅ , where the first inclusion follows from property (4) and the last equality from (3).Thus, A E ∩ A F is a closed subset of the compact space S κ \ U , hence compact.It follows from [2, 2.6d] that whenever A and B are distinct elements in Φ( E ) then A ∗ = A \ A and B ∗ are disjoint non-empty clopen subsets of ( S κ \ { p } ) ∗ . Since Φ( E ) has cardinality κ the result follows. (cid:3) In fact, if X is any compact zero-dimensional F κ -space with the property κ = κ <κ , and p ∈ X is a P κ -point of character κ then the above methods show that ( X \ { p } ) ∗ has cardinality at least κ , and contains a family of κ many disjointclopen sets. OMPACTIFICATIONS OF ω ∗ \ { x } AND S κ \ { x } Open Questions
We list open questions that are motivated by results in this paper. Corollary 6.6shows that it is consistent with CH that all remainders of the form ( ω ∗ \ { x } ) ∗ arehomeomorphic. Is the negation of this result also consistent with CH? Question 8.1.
Is it consistent with CH that for a P -point p and a non- P -point x of ω ∗ the remainders of ω ∗ \ { p } and ω ∗ \ { x } are non-homeomorphic? Theorem 1.2 shows that it is consistent with ZFC that every space ω ∗ \ { x } has a one-point Stone-Čech remainder. Under CH, Theorem 7.4 shows that theremainder of ω ∗ \ { x } has size ω for every x . Question 8.2 (van Mill) . Which cardinalities, apart from and ω , can ( ω ∗ \{ x } ) ∗ consistently have? Question 8.3.
Is it consistent that ( ω ∗ \ { x } ) ∗ and ( ω ∗ \ { y } ) ∗ have differentcardinalities for different x and y ? In Section 5 we mention the question whether one can prove without CH that ω ∗ \ { x } is a strongly zero-dimensional F -space. Question 8.4.
Is it a ZFC theorem that β ( ω ∗ \ { x } ) is a Parovičenko space? An affirmative answer to Question 8.4 proves that ( ω ∗ \ { x } ) ∗ is a compact F -space and hence rules out infinite cardinalities smaller than c in Question 8.2.Lastly, Theorem 6.3 gives us a large class of topologically distinct spaces whoseremainders are κ + -Parovičenko spaces. Can we find a precise description of spaceswith that behaviour? Question 8.5.
Is there a characterisation for which spaces X its remainder X ∗ isa κ + -Parovičenko space? References [1] W.W. Comfort and S. Negrepontis,
Homeomorphs of three subspaces of β N \ N , Math. Z. (1968), 53–58.[2] W.W. Comfort and S. Negrepontis, The Theory of Ultrafilters , Springer-Verlag, Berlin, 1974.[3] E.K. van Douwen, K. Kunen and J. van Mill,
There can be C ∗ -embedded dense proper sub-spaces in βω − ω , Proc. Amer. Math. Soc. (1989), no. 2, 462–470.[4] E.K. van Douwen and J. van Mill, Parovičenko’s characterization of βω − ω implies CH ,Proc. Amer. Math. Soc. (1978), no. 3, 539–541.[5] A. Dow, Saturated Boolean algebras and Stone spaces , Topology Appl. (1985), 193–207.[6] A. Dow, CH and Open Subspaces of F-Spaces , Proc. Amer. Math. Soc. (1983), no. 2,341–345.[7] R. Engelking, General topology , second ed., Sigma Series in Pure Mathematics, vol. 6, Hel-dermann Verlag, Berlin, 1989.[8] N.J. Fine and L. Gillman,
Extension of continuous functions in β N , Bull. Amer. Math. Soc. (1960), 376–381.[9] Z. Frolík: Sums of ultrafilters , Bull. Amer. Math. Soc. (1967), 87–91.[10] L. Gillman, The space β N and the continuum hypothesis , in: General Topology and itsRelations to Modern Analysis and Algebra, Proceedings of the second Prague topologicalsymposium. Praha, 1967. 144–146.[11] L. Gillman and M. Jerison, Rings of Continuous Functions , Springer-Verlag, New York, 1976. [12] J.F. Kennison, m -Pseudocompactness , Trans. Amer. Math. Soc. (1962), 436–442.[13] K. Kunen, Set Theory , Elsevier Science, North Holland, 1980.[14] J. van Mill,
An Introduction to βω , in: Handbook of Set-Theoretic Topology. Eds. K. Kunenand J.E. Vaughan. Elsevier Science, 1984. 503–567.[15] S. Negrepontis, The Stone space of the saturated Boolean algebras , Trans. Amer. Math. Soc. (1969), 515–527.[16] I.I. Parovičenko,
A universal bicompact of weight ℵ , Soviet Math. Dokl. (1963), 592–595.(Pitz) Mathematical Institute, University of Oxford, Oxford OX2 6GG, UnitedKingdom
E-mail address , Corresponding author: [email protected] (Suabedissen)
Mathematical Institute, University of Oxford, Oxford OX2 6GG,United Kingdom
E-mail address ::