On remote points, non-normality and π -weight ω

On remote points, non-normality and π -weight ω

Sergei Logunov

Abstract. We show, in particular, that every remote point ofXis a nonnormality point of βX ifX is a locally compact Lindel¨of separable space without isolated points and πw(X)≤ω₁.

Keywords: remote point, butterfly-point, nonnormality point Classification: 54D35

1. Introduction

We investigate some types of points in remaindersX^∗=βX\Xof ˇCech-Stone compactifications.

A pointp∈X^∗ is called a remote point ofX if it is not in the closure of any nowhere dense subset ofX. This kind of points became popular after the papers [3], [4] of van Douwen had been published. The existence of remote points in the remainders of ccc nonpseudocompact spaces withπ-weightω1 was proved by Dow [2]. An inspection of the relevant results in the literature reveals that the remote points constructed so far satisfy our condition (∗) below. This leads us to the notion of a strong remote point. It is unknown to the author whether there is an example of a remote point, which is not a strong remote point.

If removing a pointp from a compact Hausdorff space results in obtaining a nonnormal subspace, then p is called a nonnormality point of the space. There are several simple proofs that, under CH, any point ofω^∗is a nonnormality point of ω^∗ ([8], [9]). “Naively”, it is known only for special points of ω^∗. If pis an accumulation point of some countable discrete subset ofω^∗, or ifpis a strongR- point, or ifpis a Kunen’s point, thenpis a nonnormality point ofω^∗ (Blaszczyk and Szymanski [1], Gryzlov [5], van Douwen, respectively). If X is a normal second countable space without isolated points, which is either locally compact or zero-dimensional, then every point of its remainder is a nonnormality point of βX ([6], [7]).

In some cases the fact that p∈ X^∗ is a strong remote point of X permits to show thatpis ab-point ofβX, i.e. that there are setsF andG⊂X^∗\ {p}which are closed inβX\ {p}, disjoint and havepas a limit point [7], [10] (see below). It easily implies thatpis a nonnormality point ofβX, i.e.βX\ {p} is not normal.

In our paper, the following results are obtained.

Theorem 1.1. Let X be a locally compact Lindel¨of separable space without isolated points and πw(X) ≤ ω₁. Then every remote point p ∈ X^∗ of X is a b-point(and, consequently, a nonnormality point)of βX.

Theorem 1.2. LetX =S

i∈ωX_i be a normal separable space without isolated points and πw(X) ≤ ω₁. Then every strong remote point p ∈ X^∗ of X is a b-point(and, consequently, a nonnormality point)of βX.

2. Proofs

We will present a proof of Theorem 1.2 below, assuming its conditions hold. By Claims 1 and 2 it is clear that Theorem 1.1 is an easy corollary to Theorem 1.2.

The set of all functions fromω to ω is denoted byω^ω. For a set U ⊂X let U^ǫ=βX\Cl_βX(X\U) ifU is open and U^∗ =Cl_βXU \X ifU is closed. A set U ⊂X^∗ is calledτ-bounded for a cardinal τ iff for anyF ⊂U, |F| < τ implies Cl_βXF ⊂ U. A π-base U for X is a set of nonempty open subsets of X with the property that each nonempty open subset ofX contains a member ofU. The π-weight ofX,πw(X), is the minimum cardinality of aπ-base forX.

Let 2^X be set of all subsets ofX. A subsetπof 2^X is calledstrong cellular if the closures of its members inX form a pairwise disjoint family. One refines a subsetσof 2^X,π > σ, ifU∩V 6=∅impliesU ⊂V for anyU ∈πandV ∈σ. If, in addition,{U ∈π:U ⊂V} is finite for everyV ∈σ, thenπfinitally refines σ, π >_{f in} σ. And, finally,π ∗-refines σ, π >_∗ σ, iff there is a finite subset δ ⊂π such thatπ\δrefinesσ.

Ifπ₀, . . . , πnare nonempty subsets of 2^X, then the collection Yn

k=0

π_k={

\n

k=0

U_k:U_k∈π_k and

\n

k=0

U_k6=∅}

is said to be theirproduct. From now onX =S

i∈ωXi is a free topological sum andπ0={X_i:i∈ω}.

Definition 2.1. A point p∈X^∗ is called a strong remote point of X iff pis a remote point ofX and

(∗) for any family of open setsW ⊂2^X the following holds: ifW > π₀ and p∈S

W^ǫ, then there is a subfamilyW^′ ⊂ W such that W^′ >_{f in} π0 and p∈(SW^′)^ǫ.

From now on a strong remote point p ∈ X^∗ is fixed. It is easy to see that p /∈ClβXXi for eachi∈ω and that (∗) is trivial if everyXiis compact.

A discrete in X countable family of nonempty open sets π ⊂2^X is called a p-chain ifπ >_{f in}π₀ andp∈S

π^ǫ. Thusπ₀ is ap-chain. Next we put [π] =\

{Cl_βX[

σ:σ⊂π is ap-chain}

1

for any p-chain π andS ={s ∈[π₀] : sis a strong remote point ofX}. We fix Y =S

i∈ωY_i, whereY_i={y_ij :j ∈ω}is a countable everywhere dense subset of X_i, and put

T={t∈[π₀] :t∈Cl_βXD for some D⊂Y, for which everyD∩Y_i is finite}.

From now on

ξ(p) ={A⊂ω:p∈([

i∈A

X_i)^ǫ}

is an ultrafilter onω. For anyf, g∈ω^ω,f <pg iff{i∈ω:f(i)< g(i)} ∈ξ(p). It is a folklore and easy to see that there are so calledξ(p)-dominant families{fα: α < τ} ⊂ω^ω having the following properties: fα<pfβ wheneverα < β < τ and for anyg∈ω^ω,g <pfαfor someα < τ. We fix one of themF={f_α:α < λ(p)}

of the smallest cardinalityλ(p). Then, obviously, λ(p) ≥ω₁. For any G ⊂ ω^ω,

|G|< λ(p) impliesg <pf for eachg∈ G and for some f ∈ω^ω.

Now for every i ∈ ω we fix a π-base U_i = {U_iα : α ∈ ω1} for Xi. For any β∈ω₁, for{U_iα :α < β} ⊂ U_i we fix a cellular refinement{V_ij(β) :j∈ω} with the following properties:

1) everyV_ij(β) is a maximal strong cellular family of nonempty open subsets ofX_i;

2) V_ij+1(β)>V_ij(β) for each j∈ω;

3) for every α < β,V_ij(i,α,β)(β)>{U_iα} for somej(i, α, β)∈ω.

We put, also,V_g(β) =S

i∈ωV_ig(i)(β) for eachg ∈ω^ω and fix ap-chainπg(β) so thatπg(β)⊂ Vg(β).

Claims 1 through 4 are easy and sometimes well-known and are left as exercises to the reader.

Claim 1. If p∈X^∗ is ab-point of βX, thenβX\ {p} is not normal.

Claim 2. Letp∈X^∗, where X is a locally compact Lindel¨of space. Then there exists a family{Xn:n∈ω}of compact regularly closed subsets of X such that {Xn:n∈ω}is a discrete inX family andp∈Cl_βXS

{Xn:n∈ω}.

Claim 3. For anyp-chainsπandσ, if π >∗ σ, then[π]⊂[σ].

Claim 4. For any finite family of p-chains{π_i}ⁿ_i=0,Q_n

i=0π_i is ap-chain refining everyπ_i.

Claim 5. For any countable family of p-chains{π_i :i∈ω} there is ap-chainπ

*-refining everyπ_i. Proof: Letσ=S

n∈ωσ(n), where σ(n) =

Yn

i=0

{U ⊂Xn: either U ∈πi or U =Xn\Cl[ πi}.

Then ClS

σ=X. SoCl_βXOp ⊂S

σ^ǫ for some neighborhood Op⊂βX. Any p-chainπsuch thatπ⊂ {Op∩U :U ∈σmeetsOp} is as required.

Claim 6. T isλ(p)-bounded.

Proof: LetF ⊂T and|F|< λ(p). For every x∈F, x∈Cl_βXS

i∈ω{y_ij ∈Y : j ≤ fx(i)} for some fx ∈ω^ω. For some f ∈ ω^ω, fx <p f for eachx∈ F. But then

Cl_βXF ⊂Cl_βX [

i∈ω

{y_ij ∈Y :j≤f(i)} ∩[π₀]⊂T.

Claim 7. S isλ(p)-bounded.

Proof: Letq∈[π₀]\S. Then there is a maximal strong cellular family of open setsW={V_ij ⊂X_i :i, j∈ω}such thatq /∈Cl_βXS

σfor anyσ⊂ W,σ >_{f in}π₀. LetF ⊂S and|F|< λ(p). Then for every x∈ F, x∈(S

i∈ω

S

j≤fx(i)Vij)^ǫ for somefx∈ω^ω. For somef ∈ω^ω,fx<pf for eachx∈F. But then

Cl_βXF ⊂Cl_βX [

i∈ω

[

j≤f(i)

V_ij ⊂βX\ {q}.

Claim 8. For any family ofp-chains{πα}α<τ, if τ < λ(p)thenT

α<τ[πα]∩T 6=∅.

Proof: For any finite ρ⊂τ we can fix a pointt(ρ)∈T so that t(ρ)∈[Y

α∈ρ

πα]⊆ \

α∈ρ

[πα].

But then the setCl_βX{t(ρ) :ρ⊂τ is finite}, which is contained inT by Claim 6, meetsT

α<τ[πα].

Claim 9. For any family of p-chains{πα}α<τ, if τ < λ(p), thenpis not isolated inT

α<τ[πα].

Proof: LetT

α<τ[πα]∩Cl_βXOp={p}for some neighborhoodOp⊂βX. Then for a p-chain π = {Op∩X_i : i ∈ ω} we have T

α<τ[πα]∩[π]∩T = ∅ in a

contradiction to Claim 8.

Claim 10. Let T

α<τ[πα]∩S = {p} for some family of p-chains {πα}α<τ of cardinalityτ < λ(p). Thenpis ab-point of βX.

Proof: For any finite ρ⊂ τ we can fix a point s(ρ) ∈ S\ {p} so that s(ρ) ∈ [Q

α∈ρπα] by [2]. But then the sets Cl_βX{s(ρ) : ρ ⊂ τ is finite} \ {p} and T

α<τ[πα]\ {p}are as required.

Below we have only to examine the case when the hypotheses of Claim 10 are wrong.

1

Claim 11. For an arbitrary neighborhood Op ⊂βX, [π_fα(β)] ⊂ Op for some fα∈ F and β∈ω₁.

Proof: Let ClβXO^′p ⊂ Op for a neighborhood O^′p ⊂ βX. As p is a strong remote point,p∈(S S

i∈ωU_i^′)^ǫ⊂O^′pfor some finiteU_i^′ ⊂ U_i. For someβ < ω₁, U_i^′ ⊂ {U_iα:α < β}for eachi∈ω. For everyUiα∈ U_i^′we can choosej(i, α, β)∈ω so thatV_ij(i,α,β)(β) >{U_iα} (see above). Let g ∈ω^ω be defined for any i ∈ω as follows: g(i) = max{j(i, α, β) : U_iα ∈ U_i^′} if U_i^′ 6=∅ and g(i) = 1 otherwise.

Then Vg(β) > S

i∈ωU_i^′ by our construction. Let, finally, fα ∈ F be chosen so thatg <pfα. But then [π_fα(β)]⊂[πg(β)]⊂Cl_βXS S

i∈ωU_i^′ ⊂Op.

Claim 12. If|F|> ω1, thenpis a b-point of βX.

Proof: For every fα ∈ F there are points tα ∈T and sα ∈S\ {p}, belonging toB_fα =T

β<ω₁[π_fα(β)] by Claims 8 and 10. Then the setsF =Cl_βX{t_α:α <

λ(p)} \ {p} and G =Cl_βX{sα : α < λ(p)} \ {p} are as required. Indeed, they havepas a limit point by Claim 11. For everyλ < γ < λ(p), fλ <p fγ clearly implies [π_fγ(β)]⊂[π_fλ(β)] for eachβ < ω₁, soB_fγ ⊂B_fλ. But then

F∩G\Bf_λ⊂ClβX{tα:α < λ} ∩ClβX{sα:α < λ} ⊂T∩S=∅.

Claim 13. If |F|=ω₁, thenpis ab-point of βX.

Proof: Let{π_fα(β) :fα∈ F, β∈ω1}be listing into the form{πγ:γ∈ω1}. By Claim 5 we can constructp-chains σγ (γ < ω1) so that σγ >∗ πγ andσγ >∗ σλ

ifλ < γ < ω1. We can fix pointstγ∈T andsγ∈S\ {p}, belonging to [σ_γ], and repeat the proof of Claim 12, using these points.

Our proof is complete.

References

[1] Blaszczyk A., Szymanski A.,Some nonnormal subspaces of the ˇCech-Stone compactifica- tions of a discrete space, in: Proc. 8-th Winter School on Abstract Analysis, Prague (1980).

[2] Dow A.,Remote points in spaces withπ-weightω₁, Fund. Math.124(1984), 197–205.

[3] van Douwen E.K., Why certain ˇCech-Stone remainders are not homogeneous, Colloq.

Math.41(1979), 45–52.

[4] van Douwen E.K.,Remote points, Dissert. Math.188(1988).

[5] Gryzlov A.A.,On the question of hereditary normality of the spaceβω\ω, Topology and Set Theory (Udmurt. Gos. Univ., Izhevsk) (1982), 61–64 (in Russian).

[6] Logunov S.,On hereditary normality of compactifications, Topology Appl.73(1996), 213–

216.

[7] Logunov S., On hereditary normality of zero-dimensional spaces, Topology Appl. 102 (2000), 53–58.

[8] van Mill J.,An easy proof thatβN\N\ {p}is non-normal, Ann. Math. Silesianea2(1984), 81–84.

[9] Rajagopalan M.,βN\N\ {p}is not normal, J. Indian Math. Soc.36(1972).

[10] Shapirovskij B.,On embedding extremely disconnected spaces in compact Hausdorff spaces, b-points and weight of pointwise normal spaces, Dokl. Akad. Nauk SSSR223(1987), 1083–

1086 (in Russian).

Department for Algebra and Topology, Udmurt State University, Krasnogeroyskaya 71, Izhevsk 426034, Russia

E-mail: [email protected]

(Received December 6, 1999,revised November 17, 2000)