On non-normality points and metrizable crowded spaces

On non-normality points and metrizable crowded spaces

Sergei Logunov

Abstract. βX− {p}is non-normal for any metrizable crowded spaceXand an arbitrary pointp∈X^∗.

Keywords: nice family,p-filter,p-ultrafilter, projection, non-normality point, butterfly- point

Classification: 54D35

1. Introduction

We investigate non-normality points in ˇCech-Stone remaindersX^∗=βX−X of metrizable spaces.

There are several simple proofs that, under CH, ω^∗− {p} is not normal for any p∈ ω^∗ [7], [8]. “Naively” it is known only for special points ofω^∗. If pis an accumulation point of some countable discrete subset ofω^∗, or ifpis a strong R-point, or ifpis aKunen’s point, thenω^∗− {p} is not normal (Blaszczyk and Szymanski [1], Gryzlov [2], van Douwen respectively).

What about realcompact crowded spaces? Is βX− {p}non-normal whenever X is realcompact and crowded and p ∈ X^∗? Probably, but we are unaware of any counterexample. On the other hand, the answer is “yes” if X is a locally compact Lindel¨of separable crowded space with πw(X) ≤ ω₁ and p is remote [5]. It is also “yes” if X is a second countable crowded space and either X is locally compact, or X is zero-dimensional, orp is remote [3], [4], [6]. Using the regular base of Arhangel’ski˘ı J. Terasawa has omitted the separability condition in the last two cases. He has obtained the affirmative answer in case if X is a metrizable crowded space and eitherX is strongly zero-dimensional orpis remote [10]. Here, introducingp-filters into this construction, we answer affirmatively for all metrizable crowded spaces.

B. Shapirovskij [9] has defined abutterfly-point (orb-point) in a spaceX. We callp∈X^∗abutterfly-point inβX, if{p}= ClF∩ClGfor someF, G⊂X^∗−{p}

with Cl (F∪G)⊂X^∗.

Theorem. LetX be a non-compact metrizable crowded space. Then any point p∈X^∗ is a butterfly-point in βX. HenceβX− {p}is not normal.

2. Proofs

From now on a spaceX is non-compact, metrizable and crowded, i.e. X has no isolated points, and p∈X^∗ is an arbitrary point. We denote by cl- and Cl- the closure operations inX andβX respectively, 3 ={0,1,2}.

Letπandσbe an arbitrary families. A setU ∈πis called amaximal member of the family π if U ( V for noV ∈ π. If members of π are mutually disjoint (with closure), thenπis called (strongly)cellular. We writeπ≺σifU ∩V 6=∅ implies U ) V for any U ∈ π and V ∈ σ. We denote by Expπ the set of subfamilies {F : F ⊂ π}. We define a projection f_σ^π from Expπ to Expσ by f_σ^πF ={V ∈σ:S

F∩V 6=∅}for everyF ∈Expπ.

A maximal locally finite cellular family of open sets is callednice. The intro- duced in [6]cellular refinement Cel (π) ={T

φ−clS

(π−φ) :φ⊂π}ofπis nice, ifπis an open locally finite cover ofX.

Let π and σ be nice families. A collection F = {F} of subfamilies F ⊆ π is called a p-filter on π, if p ∈ Cl∪Tn

k=0F_k for any finite subcollection {F₀, . . . , Fn} ⊂ F. Obviously, the union of any increasing family of p-filters is also ap-filter. So by Zorn’s lemma there are maximalp-filters or p-ultrafilters F^′ on π, that is F^′=G for anyp-filterG withF^′⊆ G. Adding step-by-step new subfamilies from Expπ− F to F, while possible, we can embed anyp-filter F into somep-ultrafilterF^′. Ifpis not a remote point, distinctp-ultrafiltersF^′ may exist. But each of them containsπ(O) ={V ∈π:V ∩O6=∅}for any neighbor- hood O of pand its image f_σ^πF ={f_σ^πF : F ∈ F} is a p-filter on σ. We write π≺_F σ, if there isF ∈ F withF ≺σ. We denoteT

F^∗=T {ClS

F:F ∈ F}.

For everyi∈Nwe fix an open locally finite coverP_iofX so that diamU ≤ ¹_i for anyU ∈ P_i and{V ∈ P_j:V ∩U 6=∅}is finite for eachj < i. Then it is easy to see that

P = [

i∈N

P_i

is a regular base of Arhangel’ski˘ı, i.e. for any pointx∈X and for any its neigh- borhoodO ⊂X there is another neighborhood O^′ ⊂ X ofx with the following properties: O^′⊂O and at most finitely many members ofP meet boothO^′ and X−O simultaneously. Moreover, for any coverπ⊂ P the family of its maximal members is a locally finite subcover ofX.

By induction (see, also, [6]) we define the families of non-empty open setsD_k andW_k⊂ P for allk∈Nas follows:

D₁= Cel (P₁).

If a nice familyD_k={U} has been constructed, then W_k={U(ν) :U ∈ D_k and ν ∈3}

is strongly cellular with clU(ν)⊂U for any its member and D_k+1= Cel (D_k∪ W_k∪ P_k+1).

By our construction, ifU, V ∈S

k∈ND_kare not disjoint, then eitherU ⊆V or U ⊇V. For any U ∈ P_k the family ˆU ={V ∈ D_k:V ∩U 6=∅} is locally finite and nice in U. For any locally finite cover π ⊂ P we denote σ(π) all maximal members of the familyS

{Uˆ :U ∈π}. Thenσ(π) is nice. Define Σ ={σ(π) :π⊂ P is a locally finite cover ofX} and putσ(ν) ={U(ν) :U ∈σ} for anyσ∈Σ andν∈3.

Lemma 1. If πis an open locally finite cover of X, thenCel (π)is nice.

Proof: Letφ⊂π. IfT

φ6=∅, thenφis finite. SoT

φand, hence,T

φ−cl (π−φ) is open.

Letφ, φ^′ ⊂π be different andU ∈φ−φ^′. ThenT

φ⊂U and T

φ^′∩U =∅, becauseU ∈π−φ^′.

Let a neighborhoodOofx∈X meet finitely many members ofπ, sayU₁, . . . , U_k. Ifφ⊂πcontains someU ∈π− {U₁, . . . , U_k}, thenT

φ⊆U ⊆X−O. SoO meets at most 2^kmembers of Cel (π).

Asπis a locally finite family of open sets,K=S

{clU−U :U ∈π}is nowhere dense. Letx /∈K and φ={U ∈π :x∈U}. ThenU /∈φ impliesx /∈clU. So x∈ T

φ−clS

(π−φ), becauseπ is conservative, and Cel (π) is maximal. Our

proof is complete.

Lemma 2. There is a well-ordered chain{σ_α:α < λ} ⊂Σandp-ultrafiltersF_α onσα with the following properties for allα < β < λandf_β^α=f_σ^σ_β^α:

(1) p /∈ClU for eachU ∈σ₀; (2) f_β^αF_α⊂ F_β;

(3) σα≺_Fα σ_β;

(4) for anyσ∈Σ− {σα:α < λ} there isα < λwith¬(σα ≺_Fα σ).

Proof: Letπbe all maximal members of the cover{U ∈ P:p /∈ClU}and let F₀ be anyp-ultrafilter onσ₀ =σ(π).

For any ordinal β assume p-ultrafilters Fα onσα ∈Σ have been constructed for allα < β. If some σ∈ Σ− {σα : α < β} satisfies the condition σα ≺_Fα σ for allα < β, then we put σ_β =σand embed thep-filterS

α<βf_β^αFα into some p-ultrafilter F_β onσ_β. Otherwise our construction is complete.

Lemma 3. T

F₀^∗⊂X^∗.

Proof: Let x ∈ X be an arbitrary point. Then F = {U ∈ σ₀ : x /∈ clU} satisfies, obviously,x /∈ClS

F andF ∈ F₀.

Lemma 4. If α < β < λ, thenT

F_β^∗⊂T F_α^∗.

Proof: There is F ∈ Fα with F ≺ σ_β by (3). For any G ∈ Fα we have G∩F ∈ Fα andG∩F ≺σ_β. But then

\F_β^∗ ⊂Clf_β^α(G∩F)⊂Cl (G∩F)⊂ClG.

Lemma 5. For any neighbourhoodO of pinβXthere isα < λwithT

F_α^∗⊂O.

Proof: Let ClO^′ ⊂ O for a neigbourhood O^′ of p and let π be all maximal members of the cover{U ∈ P :U∩O^′6=∅ ⇒U ⊂O}. Forσ=σ(π) there isα < λ with¬(σα ≺_Fα σ) by (3) or (4). As σα(O^′)∈ Fα thenF ={V ∈σα(O^′) :V ⊆ U for someU ∈σ} also belongsFα. SoT

F_α^∗⊂ClS

F ⊂ClS

σ(O^′)⊂ClO.

Proposition 6. For anyα < λ and ν ∈ 3 there is a point pα(ν)∈ T

F_α^∗ such thatpα(ν)∈ClS

σ_β(ν)for allβ ∈λ−α.

Proof: Letα < β₀ < . . . < βn < λ be any finite sequence and F ∈ Fα. Our idea is to find non-emptyW ∈S

i≤nσ_βi so that W(ν)⊆ \

i≤n

[σ_βi(ν)∩[ F.

At the first step of induction we put ∆₀ ={σ_βi : i≤n}, Θ₀ =∅ and choose W₀∈S

∆₀as follows: We may assumeF ≺σ_β0. For anyi < nthere isG_i ∈ F_βi withGi≺σ_βi+1. We denote F₀ =f_β^α₀F ∩G₀ andF_i+1=f_β^βi

i+1Fi∩G_i+1. Then F_i+1 ≻F_i and S

F_i+1 ⊆S

F_i. Any pairwise intersecting U_i ∈ F_i make up an embedded sequenceUn⊆. . .⊆U₀⊆S

F. We defineW₀=U₀. For anym < n let ∆m,Θm ⊂∆₀ and Wm ∈S

∆m has been constructed so that

(1) ∆m∩Θm=∅;

(2) ∆m∪Θm= ∆₀; (3) Wm⊆S

F; (4) Wm⊆S

σ(ν) for anyσ∈Θm;

(5) for anyσ∈∆m there isUσ∈σwithUσ ⊆Wm. Let Ωm={σ∈∆m :Uσ =Wm}.

If ∆m6= Ωm, then we put ∆_m+1= ∆m−Ωm and Θ_m+1= Θm∪Ωm. Asσ∈

∆_m+1 are nice, we can chooseU_σ^′ ∈σso thatT

{U_σ^′ :σ∈∆_m+1} ∩Wm(ν)6=∅.

ThenUσ (Wm implies U_σ^′ ⊆Wm(ν) by our construction. We define W_m+1 to be the maximal member of embedded sequence{U_σ^′ :σ∈∆_m+1}.

If, finally, ∆m = Ωm, thenWm is as required.

Proof of Theorem: Define Fν = {pα(ν) : α < λ} for all ν ∈ 3. By our construction,Fν ⊂T

F₀^∗⊂X^∗ and for any neighbourhoodO ofpthere isα < λ with {p_β(ν) : β ∈ λ−α} ⊂ T

F_α^∗ ⊂O. Then the condition {p_β(ν) : β < α} ⊂ ClS

σα(ν) implies that the sets ClFν− {p}are pairwise disjoint andp∈Fν for no more then one uniqueFν. The other two ensure thatpis ab-point inβX.

Our proof is complete.

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Department for Algebra and Topology, Udmurtia State University, Universitetskaya 1, Izhevsk 426034, Russia

E-mail: [email protected]

(Received June 18, 2005,revised May 22, 2007)