On the structure of compact topological spaces

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On the structure of compact topological spaces

Dissertation submitted to

The Hungarian Academy of Sciences for the degree "MTA Doktora"

Zoltán Szentmiklóssy

Eötvös Loránd University Budapest

2015

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Introduction

This thesis is about some properties of compact and related topological spaces.

In the first section we study discrete subspaces of topological spaces. For example in aT2space the points of an ordinary (ωtype) convergent sequence are discrete. Some kind of spaces (metric,M₁spaces, ...) are fully determined by the convergent sequences.

But there are compact spaces, for exampleβN, which have no nontrivial convergentω- sequences. It is well known that in a compact space every non-isolated point is the limit point of a convergent transfinite sequence. For a convergentω-sequence it is also true that any initial segment doesn’t accumulate to the limit point. If we generalize this property to transfinite convergence, we get the concept of a "free sequence". We prove that if a compact space has tightness greater or equal to a regular cardinalκ > ω, then the space has a convergentκ-type free sequence.

A convergent (free) sequence in some sense has thin closure. Therefore it is an interesting problem how large can be the closure of a discrete subspace. We prove that if a weak version of GCH holds, in any countably tight compactumXthere is a discrete subspaceDwith|D|=|X|.

We also prove that if a countably compact space is the union of countably manyD subspaces then it is compact, and if a compact Hausdorff space is the union of fewer thanN(R)=cov(M)left-separated subspaces then it is scattered.

Next we study thed-separable spaces, that is the spaces which haveσ-discrete dense subspaces and we answer several problems raised by V. V. Tkachuk.

We prove the following result: If in a compact T1 space X there is a λ-branching family of closed sets thenXcannot be covered by fewer thanλmany discrete subspaces.

(A family of sets F is λ-branching iff |F | < λ but one can form λ many pairwise disjoint intersections of subfamilies ofF.) As a consequence, we obtain the following strengthening of the well-known ˇCech-Pospišil theorem: If X a is compact T₂ space such that all pointsx ∈ X have character χ(x, X) ≥ κ thenX cannot be covered by fewer than2^κmany discrete subspaces.

We call a topological space κ-compact if every subset of size κ has a complete accumulation point in it. Let Φ(µ, κ, λ) denote the following statement: µ < κ <

λ = cf(λ) and there is {Sξ : ξ < λ} ⊂ [κ]^µ such that |{ξ : |Sξ ∩A| = µ}| < λ whenever A ∈ [κ]^<κ. We show that if Φ(µ, κ, λ) holds and the space X is both µ- compact and λ-compact then X is κ-compact as well. Moreover, from PCF theory we deduce Φ(cf(κ), κ, κ⁺) for every singular cardinal κ. As a corollary we get that a linearly Lindelöf andℵ_ω-compact space is uncountably compact, that isκ-compact for all uncountable cardinalsκ.

The next section is about the relation of calibers and density. We prove, among others, the following theorems:

– IfXis aT₃ space with no free sequences of lengthλ and(λ, λ, κ)is a caliber of

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Xthend(X)≤µ^<λfor some cardinalµ < κ.

– IfX isT3andX =S

C with|C|< κand the members ofC are compact with no free sequences of lengthµ, moreover(µ, κ)is a caliber ofXthend(X)< κ.

– IfX isT₃ andX =S

C with|C| ≤ κ andCis compact with no free sequences of lengthκfor everyC ∈ C, moreoverκis a caliber ofX thend(X)< κ.

These results provide strengthenings and generalizations of some results of Šapirovskii and of Arhangelskii, respectively.

We define the projectiveπ-characterp πχ(X)of a spaceX as the supremum of the valuesπχ(Y)whereY ranges over all (Tychonov) continuous images ofX. Our main result says that every Tychonov spaceXhas aπ-base whose order is≤p πχ(X), that is every point inX is contained in at mostp πχ(X)-many members of theπ-base. Since p πχ(X) ≤ t(X) for compact X, this is a significant generalization of a celebrated result of Šhapirovskii.

We answer several questions of V. Tkachuk from [Point-countable π-bases in first countable and similar spaces, Fund. Math. 186 (2005), pp. 55–69.] by showing the following results:

– There is a ZFC example of a first countable, 0-dimensional Hausdorff space with no point-countable π-base (in fact, the minimum order of a π-base of the space can be made arbitrarily large).

– If there is a κ-Suslin line then there is a first countable GO space of cardinality κ⁺in which the order of anyπ-base is at leastκ.

– It is consistent to have a first countable, hereditarily Lindelöf regular space having uncountableπ-weight andω₁ as a caliber (of course, such a space cannot have a point-countableπ-base).

The final section is about characterizing continuity by preserving compactness and con- nectedness. Let us call a function f from a space X into a space Y preservingif the image of every compact subspace of X is compact in Y and the image of every connected subspace ofXis connected inY. By elementary theorems a continuous function is always preserving. Evelyn R. McMillan [46] proved in 1970 that ifX is Hausdorff, locally connected and Frèchet,Y is Hausdorff, then the converse is also true: any preserving functionf : X → Y is continuous. The main result of this part is that ifX is any product of connected linearly ordered spaces (e.g. ifX =R^κ) andf :X → Y is a preserving function into a regular spaceY, thenf is continuous.

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Notations

We use all the standard notations of set theory.

• IfI is an ideal on the setX such that[X]^<ω ⊂ I then thecovering number ofI defined ascov(I) = min{|J |:J ⊂ I és [

J } =X

• If M is the ideal on R generated by the nowhere dense subsets then N(R) = cov(M)denotes theNovak-numberof the real line.

• s= min{|A|:Asplitting family}thesplitting number.

A ⊂[ω]^ω is asplitting familyif∀Y ∈[ω]^ω∃A∈ A |Y ∩A|=|Y \A|=ω

• p= min{|F |:F ⊂[ω]^ωstrongly centered and∀A∈[ω]^ω∃F ∈ F |A\F|=ω}

F ⊂[ω]^ωstrongly centeredif∀f ∈[F]^<ω| ∩f|=ω

• IfB ⊂P(X)then theorder ofB

ord(B) = sup{ord(x,B) :x∈X} whereord(x,B) =|{B ∈ B :x∈B}|

The familyBispoint-countableiford(B)≤ω Cardinal functions

In this dissertation we assume that all topological spaces are infiniteT1spaces.τdenotes the open sets ofXandτ^∗ =τ \ {∅}the family of the nonempty open sets. Now letX be a topological space andx∈Xan arbitrary point in the space.

• w(X) = min{|B| :Bis a base ofX}is theweightofX.

The spaceXis said to be anM2 space ifw(X) = ω.

• π(X) = min{|B|:Bis aπ-base}is theπ-weightofX.

• d(X) = min{|S|:S =X}is thedensityofX.

• L(X) = min{κ : ∀Gopen cover∃G⁰ ∈ [G]^≤κsubcover} is the a Lindelöf- numberofX.

The spaceXsaid to beLindelöfifL(X) =ω.

• c(X) = sup{|C|:C ⊂ τ is cellular}is thecellularityofX.

Here "cellular" means "pairwise disjoint".

The spaceXhas theSuslin-property (shortly Suslin)ifc(X) = ω.

• s(X) = sup{|D|:D⊂Xis discrete}is thespreadofX.

A spaceS isleft (right) separatedif there is a well-ordering ofS such that every final (initial) segment is open.

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• z(X) =hd(X) = sup{|S|:Sis left separeted}.

• h(X) =hL(X) = sup{|S|:Sis right separeted}.

{x_α :α∈κ} ⊂Xis afree sequenceif∀α ∈κ{x_β :β < α} ∩ {x_β :β≥α}=∅.

• F(X) = sup{|S|:Sis a free sequence}.

Some of the (global) cardinal functions (likes(X), z(X), h(X), F(X)) are defined in the form:

ϕ(X) = sup{|Y|:Y ⊂Xandψ(Y)}

whereψis a property of spaces that is inherited by subspaces. In this case we can define the "bversion" ofϕas follows:

ϕ(X) = min{|Yb |:Y ⊂Xand¬ψ(Y)}

In this way we get the cardinal functionsbs(X), bz(X), bh(X), Fb(X).

• χ(x, X) = min{|B| : B ⊂ τ is a neighbourhood base ofx}is thecharacter ofx inX.

χ(X) = sup{χ(x, X) :x∈X}is thecharacterofX.

The spaceXis said to be anM₁ space orfirst-countablespace ifχ(X)≤ω.

• ψ(x, X) = min{|B|:B ⊂ τ,T

B={x}}is thepseudo-character ofxinX ψ(X) = sup{ψ(x, X) :x∈X}is thepseudo-characterofX.

• t(x, X) = min{κ : ∀A ⊂ X, x ∈ clA =⇒ ∃B ⊂ A,|B| ≤ κ, x ∈ B}is the tightness ofxinX.

t(X) = sup{t(x, X) :x∈X}is thetightnessofX.

• πχ(x, X) = min{|B|:Blocalπ-base ofx-ben}is theπ-character ofxinX.

πχ(X) = sup{πχ(x, X) :x∈X}is theπ-characterofX.

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1 Discrete subspaces

1.1 Convergent free sequences in compact spaces

While there are compact T₂ spaces with no nontrivial convergent ω-sequences, βN being perhaps the best known such space, every infinite compact T₂ space contains nontrivial convergent transfinite sequences. Indeed, as it is easy to see, if in such a space{A_α :α∈ ϕ}is a (strictly) decreasing sequence of closed sets withT{A_α :α ∈ ϕ} = p, a singleton, then for every choice of points q_α ∈ A_α \ A_α+1, the sequence {qα :α ∈ϕ}convergences top, provided thatϕis a limit ordinal.

Moreover, it is also easy to see that if the character of a point pin such a space is equal toκthen a decreasing sequence of closed sets{A_α :α ∈κ}does exist with

\{A_α :α∈κ}=p,

hence p is the limit of a κ-sequence. Consequently, since every infinite compact T₂ space has a separable closed infinite subspace, in which then every point has character

≤c= 2^ω, we get that every such space has convergent sequences of length≤c.

On the other hand, it has also been known (see [8] or [71]) that for example, βN contains a convergent sequence of lengthω₁ , without assuming CH, hence the natural question arises whether every infinite compactT₂ space contains a convergentω orω₁, sequence? This question was first formulated by Hušek in the late seventies, and a related stronger problem was independently raised by István Juhász around the same time:

Does every nonfirst countable compactT₂space contain a convergentω₁ sequence?

In this section we show that every compactT₂ space of uncountable tightness contains a convergentω1 sequence, moreover assuming CH every nonfirst countable com- pactT₂space does. Also we obtain that under CH every compactT₂ space with a small diagonal is metrizable, thus solving another problem of Husek from [24]. More gener- ally, using some results of Dow from [12] we obtain that these consequences will hold in any generic extension obtained by adding Cohen reals to any ground model satisfying CH.

1.1.1 The Main theorem

The main result of this section says that if κ is any uncountable regular cardinal and X is a compact T₂ space containing a free sequence of length κ, thenX also contains a convergent such sequence. Let us note that having a free sequence {p_α : α ∈ κ}

converging to p in X also yields a closed set F ⊂ X with χ(p, F) = κ. Indeed, F ={p_α :α∈κ}works since the sets{p_α :α∈κ\β}are clopen inF for allβ ∈κ, for the sequence{p_α :α ∈κ}is free, i.e.,

{p_α :α∈β} ∩ {p_α :α∈κ\β}=∅

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forβ ∈κ.

The proof of the main result is split into two cases according to whetherXcontains a closed set that maps continuously onto 2^κ or not. In the first case we get somewhat stronger results and now we turn to their discussion.

Theorem 1.1. Letκ be any uncountable cardinal andf : X → 2^κ be an irreducible continuous map of the compactT₂ spaceX onto2^κ . Then for every point x ∈ X we have

(i) there is a (relatively) discrete subspace D ⊂ X \ {x} with |D| = κ that "converges" tox in the strong sense that every neighbourhood of x contains all but countably many elements ofD;

(ii) moreover, ifcf(κ)> ω, then there is a free sequence{xα :α∈κ} ⊂X\ {x}that converges tox.

Proof.We need several simple facts concerning irreducible maps that are probably well known, although we have not found them in the literature. For the sake of completeness we present them here with their proofs. First, the image of a regular closed set (in particular, of a clopen set)A under an irreducible closed map f : X → Z is regular closed. Indeed, if we hady∈f(A)\intf(A)6=∅thenA\f⁻¹

intf(A) , hence G= intA\f⁻¹

intf(A) 6=∅

becauseAis regular closed. Sincef is irreducible we haveintf(G)6=∅, contradicting that bothintf(G)\intf(A)andf(G)∩intf(A) = ∅.

Next we show that iff is as above andA, Bare regular closed sets inX, then intf(A∩B) = intf(A)∩intf(B).

Indeed, assume thatintf(A∩B)$intf(A)∩intf(B). Then

C = (intf(A)∩intf(B))\f(A∩B) 6=∅, henceD= intA∩f⁻¹(C)6= ∅, as well.

But thenD∩B =∅andf(D)\C\f(B); consequently,f(X\D) =Y, contradicting thatf is irreducible.

Let us now return to the proof of the theorem, where, for technical reasons, we first assume thatXis also0-dimensional.

For S ⊂ κ a subset H of 2^κ is called S-determined if z ∈ H implies z⁰ ∈ H whenever z⁰ ∈ 2^κ and z⁰ S = z S. It is well known (cf. [15]) that every regular closed (or open) subset of2^κ isS-determined for some countableS ⊂ κ. Moreover, it is obvious that ifH isS-determined then so areH andintHas well.

Forα ∈κandi∈2we shall put

U_α,i ={z ∈2^κ :z(α) = i}

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so theU_α,iform the canonical subbase for the product topology on2^κ.

Now, for any subset S ⊂ κwe let A_S be the collection of all those clopen subsets A of X that contain x and whose f-image f(A) is S-determined. According to our preliminary remarks, for every clopen neighbourhood A of xthere is some countable set S ⊂ κ such that A ∈ A_S . Moreover, each collection A_S is closed under finite intersections. Finally, we set TA_S = F_S. Then F_S is a closed set containing x, and clearlyS ⊂T impliesF_S ⊂F_T.

Let us putf(x) =yand for anyS ⊂κwe set

Φ_S ={z ∈2^κ :z S =yS}.

We claim thatf(F_S) = Φ_S holds for everyS ⊂κ. Indeed, first note that f(F_S) =f(\

A_S) =\

{f(A) :A∈ A_S}

becauseXis compact andA_Sis closed under finite intersections. But for allA ∈ A_S we havey∈f(A)andf(A)isS-determined, henceΦ_S ⊂f(A), i.e.,Φ_S ⊂f(F_S). On the other hand, for eachα∈S we clearly havef⁻¹(U_α,y(α))∈ A_S. Consequently,

f(F_S)⊂\

{U_α,y(α) :α ∈S}= Φ_S hencef(F_S) = Φ_S.

Now for everyα∈κwe lety^αbe the point of2^κ that agrees withyforβ ∈κ\ {α}

but disagrees with it atα, i.e.,

y^α(β) =







y(β) ifβ 6=α 1−y(α) ifβ =α

.

SinceΦ_κ\{α} = {y, y_α, we may pick the pointsx^α ∈ F_κ\{α} such thatf(x^α) = y^α for allα∈κand claim that the setD={x^α :α∈κ}satisfies (i).

That Dis discrete follows immediately from the fact that the set {y^α : α ∈ κ} is discrete. Next, ifAis any clopen neighbourhood ofxthen there is a countable setS ⊂κ withA∈ A :S. But then for everyα∈κ\S, we have

x^α∈F_κ\{α} ⊂F_S ⊂A, hence (i) holds because X is0-dimensional.

As for (ii), let us first define the pointsyα ∈2^κ forα ∈κby the stipulation

y_α(β) =







y(β) ifβ ≤α 1−y(β) ifα < β < κ

.

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Clearly {y_α : α ∈ κ} is a free sequence in 2^κ. Moreover, y_α ∈ Φ_α+1 for each α ∈ κ. Sincef(F_α+1) = Φ_α+1, we can choose pointsx_α ∈ F_α+1 forα ∈ κsuch that f(x_α) = y_α.

Now {x_α : α ∈ κ} is a free sequence in X: if z ∈ {x_α :α∈β} then f(z) ∈ {y_α :α∈β} by the continuity of f; hence f(z) ∈ {y/ _α :α ∈β\β} because {y_α} is free; thusz /∈ {x_α :α∈β\β}.

Finally to see that {x_α : α ∈ κ} converges to x it clearly suffices to show that T{F_α+1 :α∈κ}={x}since theF_α+1 are decreasing andx_α ∈F_α+1\F_α+2. But for every clopen neighbourhoodAofxthere is a countableS ⊂κwithA ∈ A_S, hence by cf(κ)> ωthere is anα∈κsuch thatS⊂α. Consequently,F_α+1 ⊂F_α ⊂F_S ⊂A.

Thus, using again thatX is0-dimensional, (ii) has been established.

Now let us consider the general case with an arbitrary compact T₂ space X. By Alexandrov’s well-known theorem (see, e.g., [15, 3.2.2]), there is a0- dimensional com- pactT₂spaceZadmitting an irreducible mapg :Z →XontoX. Then the composition h =f ◦g : Z → 2^κ is also irreducible, hence the above considerations can be applied toh and a fixed pointz ∈ Z with g(z) = x. This gives us points{z^α : α ∈ κ}with h(z^α) = y^α and{z_α : α ∈ κwithh(z_α) = y_α. Now it is straightforward to check that the setD ={x^α =g(z^α) :α ∈ κ}and the free sequence{x_α =g(z_α) : α∈ κ}are as required by (i) and (ii), respectively.

Now we turn to the formulation of our main result.

Theorem 1.2. Let κ be an uncountable regular cardinal. If a compact T₂ space X contains a free sequenceS ={x_α :α ∈κ}of lengthκthen it also contains one that is convergent.

Proof. Let us first note that the closure S of S can be mapped continuously onto the spaceκ+ 1taken with its usual order topology. Indeed, if for any limit ordinalλ < κ we set

A_λ =\

{x_β :β ∈λ\α}, or equivalently, sinceS is free,

A_λ ={x∈X :λ= min{α:x∈ {x_β :β ∈α}⁰}}, then the mapf :S →κ+ 1(defined below) is clearly continuous:

f(x) =







n x=x_n, n < ω α+ 1 x=x_α, α∈κ\ω

λ x∈A_λ, λ≤κlimit

Thus we may assume without any loss of generality that X admits an irreducible mapfontoκ+ 1. Moreover, using1.1(ii), we may also assume that no closed subspace

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ofX can be mapped onto2^κ. This, however, by Sapirovskil’s celebrated result (cf. [25, 3.18]) implies that every nonempty closed subsetF ofX has a pointpwithπχ(p, F)<

κ.

For every ordinalα < κlet us defineZ_α =f⁻¹((κ+ 1)\α). In particular, then the setZ_κ =f⁻¹({κ})is nowhere dense inXbecausef is irreducible.

Letp ∈ Z_κ be any point havingπ-characterπχ(p, Z_κ) = π less thanκ inZ_κ. We are going to show that there is a free sequence of lengthκ converging top. This will clearly establish our result.

First we claim that for every open subset GofX ifG∩Z_κ 6= ∅thenf(G)∩κis cofinal inκ. Indeed, otherwise we had anα ∈ κsuch thatG\Z_κ ⊂ f^−l[α], hence as Z_κ is nowhere dense we had

G⊂G\Z_κ ⊂f⁻¹[α]⊂f⁻¹[α+ 1], contradicting thatG∩Z_κ 6=∅.

Usingπχ(p, Z_κ) =π(< κ), we may choose open sets{G_γ :γ ∈π}inX such that {G_γ∩Z_κ : γ ∈ π}forms a localπ-base atpinZ_κ. Let us pick a pointq_γ ∈ G_γ ∩Z_κ for eachγ ∈ π and then choose the open setB_γ inX such thatq_γ ∈ B_γ ⊂ B_γ ⊂ G_γ. By the above claim, then

C_γ =κ∩f(B_γ)

is a closed unbounded subset ofκ, hence byπ < κso isC = T{C_γ : γ ∈ π}. (Here we use the regularity ofκ.)

Now, for everyα ∈ κwe letV_α be the collection of all those open neighbourhoods V ofpfor which B_γ∩Z_κ ⊂ V impliesB_γ∩Z_α for allγ ∈ π. Clearly V₁, V₂ ∈ V_α impliesV₁ ∩V₂ ∈ V_κ. MoreoverV_α ⊂ V_β ifα < β. Let us put

Fα =\

{V :V ∈ Vα}

Then we haveF_α ⊃F_β forα < β and f(F_α) =\

{f(V) :V ∈ V_α} by our above remarks.

But for every V ∈ V_α, there is a γ ∈ π such that B_γ ∩ Z_α ⊂ G_γ ∩ Z_α ⊂ V. Consequently,

C\α⊂C_γ\α⊂f(B_γ∩Z_α)⊂f(V).

Thus we obtainC\α⊂f(F_α)and, in particular,F_α6={p}.

Next we show that for every neighbourhood V ofpthere is someα ∈ κ withV ∈ V_α. Indeed, ifB_γ∩Z_κ ⊂V then

Z_κ =\

{Z_α :α∈κ},

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andZ_α ⊂ Z_β forβ < αimply that, asX is compact, there is someα_γ ∈ κ such that B_γ ∩Z_α_γ ⊂ V. Thus if a α ∈ κ is chosen with α_γ ≤ α for each γ ∈ π, and this is possible sinceκis regular andπ < κ, thenV ∈ V_αindeed.

Putting all this together we obtain from the setsF_αa strictly decreasing sequence of closed sets whose intersection is{p} . By suitably thinning out, we may assume that F_α+1 $ F_α for all α ∈ κ; hence ifp_α ∈ F_α+1 \F_α then the sequence{p_α : α ∈ κ}

converges top. Finally, by passing to a subsequence we may assume that ifα < βthen f(p_alpha) < f(p_β), and then the subsequence{p_α+1 : α ∈ κ}is also free because its f-image{f(p_α+1) :α∈κ}is.

1.1.2 Applications

In this section we are going to present several results that are more or less immediate consequences of our main theorem and other known results. To start with, we consider a result which, forκ=ω, was formulated in the introduction.

Corollary 1.3. If2^κ =κ⁺ andX is a compactT2 space of characterχ(X) > κ, then X has a closed subsetF with a point p such that χ(p, F) = κ⁺, hence a convergent κ⁺-sequence as well.

Proof.Ift(X) > κthen this is immediate from 1.2, even without assuming2^κ = κ⁺. (Remember that a convergent free κ⁺-sequence yields F and p as required.) Now if t(X) ≤ κthen for any pointp ∈ X withχ(p, X)≥ κ⁺ we may apply 6.14(b) of [25]

withλ =κ⁺to obtain a setY ⊂Xwith|Y| ≤κ⁺such that χ(p, Y) = χ(p, Y)≥κ⁺.

If|Y| ≤κthen we havew(Y)<2^d(Y⁾ ≤2^κ =κ⁺, hence we must haveχ(p, Y) = κ⁺. If, on the other hand,|Y|=κ⁺then let us writeY =S

{Y_α:α ∈κ⁺}with|Y_α|=κ andY_α ⊂ Y_β forα < β < κ⁺. Since t(X) ≤ κ, we also have Y = S

{Y_α : α ∈ κ}.

But then for eachα ∈κ⁺we havew(Y_α)< κ⁺as above, hence (cf. [25]) we have w(Y) = nw(Y)≤X

{nw(Y_α) :α∈κ⁺} ≤κ⁺, and thus againχ(p, Y) =κ⁺.

For the caseκ=ωwe can prove the following partial strengthening of1.3.

Corollary 1.4. IfV satisfies CH andW is an extension ofV obtained by adding some Cohen reals toV, then inW every nonfirst countable compactT₂ spaceX has a con- vergentω₁-sequence. (Note thatW =V is permitted here.)

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Proof. As above, using our main theorem we can assume that t(X) = ω, hence, by 5.8(b) of [12], every pointp∈Xwithχ(p, X)> ωis the limit of anω₁-sequence.

In [24] Hušek introduced the notion of small diagonal and asked whether CH implies that every compactT₂space with a small diagonal is metrizable. (Recall thatXis said to have small diagonal if for every uncountable setH ⊂X²\∆there is a neighbourhood U of∆withH\U also uncountable. In other words, this means that noω₁-sequence fromX²\∆can converge to∆.) We can now establish the following stronger result.

Corollary 1.5. If W is as in 1.4 then in W every compact T₂ space X with a small diagonal is metrizable.

Proof.By 5.8(c) of [12] this is so ift(X) ≤ ω. If on the other handt(X) > ω, then by 1.2 there is a convergent ω₁-sequence {p_α : α ∈ ω₁} in X consisting of distinct points. But ifhp_αiconverges topthen{(p_α, p_α+1) : α ∈ ω₁} ⊂ X²\∆converges to (p, p)∈∆, which shows thatX cannot have small diagonal.

Let us now recall from the introduction Hu’sek’s other conjecture saying that every compactT₂ space contains a convergentω- orω₁-sequence. Our next result shows that this is implied by a conjecture formulated by István Juhász in [28], namely, that every compactT2 space of countable tightness has a point of character≤ ω1. In fact, a little more is true.

Corollary 1.6. If every compactT₂space of countable tightness has a point of character

≤ω₁then in every infinite compactT₂spaceXthere is a closed setF with a pointp∈F such thatχ(p, F) =ωorχ(p, F) =ω₁.

Proof. Ift(X) > ω then we can simply apply 1.2. Next if t(X) = ω and X is also scattered, then X contains the one-point compactification of an infinite set, hence a closed setF with a pointpsatisfyingχ(p, F) = ωorχ(p, F) = ω₁. Finally ift(X) =ω andX is not scattered, thenX contains a closed setF that is dense in itself and, by our assumption, for some pointp∈F we haveχ(p, F) = ωorχ(p, F) = ω₁.

In [29] it was shown that the assumption of1.6 is valid in a generic extension obtained by addingω1 Cohen reals to an arbitrary ground model. Moreover, Dow proved in [13] that this assumption, hence by1.6 also Hušek’s conjecture, is valid under PFA.

Next we give a very simple and quick proof of a result form [30]. In order to formulate this we recall that a space X is said to omit a cardinal λ if|X| > λ butX does not contain a closed subset of sizeλ.

Corollary 1.7. If 2²^κ =κ⁺⁺then no compact spaceX may omit bothκ⁺andκ⁺⁺. Proof.Let|X| > κ⁺⁺. If t(X) ≤ κ then for everyY ⊂ X with|Y| = κ⁺⁺ we have

|Y|=κ⁺⁺because

Y =[

{Z :Z ⊂Y and|Z| ≤κ}.

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If, however,t(X) > κ, then by1.2 there is a convergentκ⁺-sequenceS = {x_α : α ∈ κ⁺}inX. Then we have

S =[

{x_β :β ∈α}:α∈κ⁺} ∪ {x}, wherexis the limit ofS, hence clearly either|S|=κ⁺or|S|=κ⁺⁺.

Note that the same proof actually yields in ZFC that no compact T2 space can omit every cardinal in the interval[κ⁺,2^κ].

We finish with a result that is not connected with our main theorem, still we thought to add it here because it is directly related to Hušek’s conjecture. This result uses the set theoretic principle♣(club), introduced by Ostaszewski in [49], saying that with every limit ordinal λ ∈ ω₁ we can associate an ω-type subset S_λ ⊂ λ withS

S_λ = λ such that for every uncountable setH ⊂ ω1 there is a λ with Sλ ⊂ H. ♣is known to be consistent with the continuum being arbitrarily large.

Theorem 1.8. Assume♣and letX be a countably compact infiniteT₂ space that contains no nontrivial convergent ω-sequence. Then there is an ω₁-size supspace Y = {x_α : α ∈ ω₁}of X such that in Y every (relatively) open set is either countable or co-countable

Proof.LethS_λ :λ∈L₁ibe a♣-sequence withL₁ denoting the set of limit ordinals in ω₁. By transfinite induction on a α ∈ ω₁, we are going to define infinite closed sets Fα ⊂X and pointsxα ∈Fα as follows.

Let us putF₀ = X, and if the infinite closed setF_α has been defined then we pick x_α ∈F_αandF_α+1 ⊂F_α\ {x_α}such thatF_α+1 be closed and infinite.

Ifλ ∈L1 and theFαhave been defined for allα∈λthenhFα :α ∈λiis a decreas- ingω-sequence of closed sets, hence the setF_λ = {x_α : α ∈ S_λ}⁰ of all accumulation points of the sequence {x_α : α ∈ S_λ}is nonempty by countable compactness. More- over, it is infinite because X contains no convergentω-sequence. This completes the induction.

It is immediate from our construction that for everyλ∈L₁we have {X_α :α∈ω₁\λ} ⊂ {x_β :β ∈S_λ}.

Consequently, since for every uncountable setZ ⊂ Y there is a λ ∈ L₁ with{x_beta : β ∈S_λ} ⊂Z, any uncountable closed (or open) subset ofY must be co-countable. The proof is thus completed.

Let us first note thatY is anS-space if it is also T3. Moreover it is obvious thatY can have at most one complete accumulation point. Consequently, we have

Corollary 1.9. If♣holds then every infinite initially ω₁-compactT₂-space has a nontrivial convergentωorω₁-sequence.

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1.2 Discrete subspaces of countably tight compacta

In this sectioncompactumwill mean an infinite compact Hausdorff space. The cardinal functiong(X)denotes the supremum of cardinalities of closures of discrete subspaces of a spaceX; this was introduced in [2]. It was also asked there if |X| = g(X)holds for every compactumX. Much later, at a meeting in Budapest in 2003, Archangelskiˇı also asked the following (slightly) stronger question: Does every compactumXcontain a discrete subspace Dwith|D| = |X|? (If, following the notation of [25], we denote bybg(X)the smallest cardinalκsuch that for every discreteD ⊂ X we have |D| < κ then this latter question asks ifbg(X) =|X|⁺for every compactumX.)

It was noted by K. Kunen that the answer to the second question is “no” if there is an inaccessible cardinal, because for every non-weakly compact inaccessible cardinalλ there is even an ordered compactum X with bg(X) = |X| = λ. Moreover A. Dow in [11] gave, with the help of a forcing argument, a consistent counterexample to the first question. It remains open if there are ZFC counterexamples to either question.

On the other hand, A. Dow proved in [11] the following positive ZFC results for countably tightcompacta.

Proposition 1.10. (A. Dow, see in [11]) LetX be a countably tight compactum. Then (i) |X| ≤g(X)^ω;

(ii) if|X| ≤ ℵ_ωthenbg(X) =|X|⁺.

A. Dow also formulated the following conjecture in [11]: For every suchX we do have bg(X) = |X|⁺. In what follows we shall confirm this conjecture under a slight weakening of GCH, namely the assumption that for any cardinal κ the power 2^κ is a finite successor ofκ; or equivalently: every limit cardinal is strong limit.

It is trivial that for any space X we have s(X) ≤ g(X) and bs(X) ≤ bg(X) , however the following stronger inequality, which we shall use later, is also true:h(X)≤ g(X) andbh(X)≤ bg(X)

. This is so because for any right separated (or equivalently:

scattered) spaceY the setI(Y) of all isolated points of Y, that is clearly discrete, is dense inY.

We start by giving a new, simpler proof of1.10(i).

Proof of1.10(i). LetX be a countably tight compactum andM be a countably closed elementary submodel of H(ϑ) for a large enough regular cardinal ϑ with X ∈ M, g(X)⊂M and|M|=g(X)^ω.

First we show thatX∩M is compact. Indeed, ifDis any countable discrete subspace of X∩M then D ∈ M becauseM is countably closed, henceD ∈ M and so g(X) ⊂ M impliesD ⊂ M. AsX is countably tight, it follows that theX-closure of any discrete subspace ofX ∩M is contained inX ∩M, hence the subspace X∩M has the property that the closure of any discrete subspace of it is compact. But then it is well-known (see e.g. [63]) thatX∩M is compact.

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Next we show that X = X ∩M; this will clearly complete the proof. Assume, indirectly, thatx∈X\M. By 2.10 (a) of [25] we haveψ(X)≤h(X)≤g(X), hence again byg(X) ⊂ M there is for each pointy ∈ X∩M a localψ-baseU_y ⊂ M. Pick for eachy∈X∩M an elementU_y ∈ U_y ⊂M withx6∈U_y. SinceX∩M is compact, from the open cover{U_y : y ∈ X ∩M} ofX ∩M we may select a finite subcover U ={U_y₁, . . . , U_y_n}. But thenU ∈M andx6∈ ∪ U, a contradiction.

Before giving our next result we shall prove two lemmas that may turn out to have independent interest. Both lemmas say something aboutT₃spaces.

Lemma 1.11. LetXbe a countably compactT₃ space andλbe a strong limit cardinal of countable cofinality such thatπχ(X) < λ, moreover|G| ≥ λ for each (non-empty) open subsetGofX. Then we actually havebg(X)> λ^ω, i. e. Xhas a discrete subspace Dwith|D| ≥λ^ω.

Proof.By a well-known result of Šapirovskiˇı (see [25], 2.37), for every open Gin X we have c(G) ≥ λ, because otherwise we had w(G) ≤ πχ(G)^c(G) < λ, hence also

|G| ≤2^w(G)< λasλis strong limit. But then by a result of Erd˝os and Tarski (see [25], 4.1) we also havebc(G) > λ, i.e. G contains λ many pairwise disjoint open subsets, becauseλis singular.

Now given an open setG⊂Xand a pointx∈Glet us fix an open neighbourhood V(G, x) =V ofxsuch thatV &G. By the above we may also fix a familyU(G, x)of open subsets ofG\V with pairwise disjoint closures and with|U(G, x)| =λ. This is possible becauseX isT₃.

We next define for all finite sequencess ∈λ^<ω open setsG_sand pointsx_s ∈G_sby recursion on|s|as follows. To start with, we setG∅ =X and pickx∅ ∈G∅ arbitrarily.

Oncex_s ∈ G_s are given, then the sets {Ga

sα : α ∈ λ}are chosen so as to enumerate U(G_s, x_s)in a one-to-one manner. Then the pointsxa

sα ∈Ga

sα are chosen arbitrarily.

Let us setD={x_s :s∈λ^<ω}. ThenDis discrete because, by the construction, we clearly haveD∩V(G_s, x_s) = {x_s}for eachs ∈ λ^<ω. For everyω-sequencef ∈ λ^ω, using the countable compactness ofX, we may choose an accumulation pointx_f of the set{x_f_n : n ∈ ω}. Clearly, we have x_f ∈ G_fn for all n ∈ ω. Forf₁, f₂ ∈ λ^ω with f₁ 6=f₂ thenx_f₁ 6=x_f₂ because ifnis minimal withf₁(n)6=f₂(n)then

G_f₁_n+1∩G_f₂_n+1 =∅.

This shows that|D| ≥λ^ω, hencebg(X)> λ^ω, and the proof is completed.

As we shall see below, using 1.10(i) and 1.11one can already prove |X| = g(X) for countably tight compacta, under the assumption that2^κ < κ^+ω for allκ. However, to get the stronger resultbg(X) = |X|⁺for the case when|X|is an inaccessible cardinal, we shall need the following lemma.

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Lemma 1.12. Letκ, λ, µbe cardinals withκ →(λ, µ)² andX be aT₃ space in which there is a left-separated subspace of cardinalityκ(i.e.bz(X)> κ). Then eitherX has a discrete subspace of sizeλ(i.e. bs(X)> λ)or there is inX a free sequence of lengthµ (i.e.Fb(X)> µ).

Proof.LetY ⊂ X be left-separated by the well-ordering≺ in order-typeκ. Then for eachy∈Y we can fix aclosedneighbourhoodN_y such that{z ∈Y :z ≺y} ∩N_y =∅.

Let us then define the coloringc: [Y]² → 2by the following stipulation: fory, z ∈ Y withy ≺ z we have c({y, z}) = 0if and only ifz 6∈ N_y. Byκ → (λ, µ)² then we either have a 0-homogeneous subset ofY of sizeλor a 1-homogeneous subset of sizeµ.

But clearly, ifS ⊂Y is 0-homogeneous thenS is discrete because thenS∩N_y ={y}

for all y ∈ S, and ifS is 1-homogeneous then S is free inX because in this case for anyy∈Swe have{z ∈S: z ≺y} ∩N_y =∅and{x∈S: yx} ⊂N_y.

We are now ready to present our result.

Theorem 1.13. Assume that 2^κ < κ^+ωholds for all cardinalsκ. Then for every countably tight compactumXwe havebg(X) =|X|⁺, i.e. there is a discrete subspaceD⊂X with|D|=|X|.

Proof. Let us first consider the case in which |X| is a limit cardinal, hence by our assumption a strong limit cardinal. If |X|is also singular then by a result Hajnal and Juhász (see [25], 4.2) we have bs(X) = |X|⁺, i.e. there is even a discrete D ⊂ X with |D| = |X|. If, on the other hand, |X| is regular, hence inaccessible, then we have d(X) = |X| (and so bz(X) = |X|⁺) because |X| is strong limit, moreover by a well-known result of partition calculus (see e.g. [16]) we have |X| → (|X|, ω₁)². But obviously in a countably tight compactum there is no free sequence of uncountable length, hence applying lemma1.12we again obtain a discreteD⊂X with|D|=|X|.

So next we may assume that|X|is a successor cardinal:|X|=λ⁺ⁿwhereλis limit andn ∈ω\ {0}.

Now, if|X|isω-inaccessible, i.e. for anyµ < |X|we have µ^ω < |X|, then by 1.1 (i) we must haveg(X) = |X|. But then,g(X)being a successor cardinal, we must also havebg(X) =|X|⁺.

If, on the other hand,|X|isω-accessible then our assumptions clearly implycf(λ) = ω and λ < |X| ≤ λ^ω. If bh(X) = |X|⁺ then by bh(X) ≤ bg(X) we also have bg(X) = |X|⁺, hence we may assume that bh(X) ≤ |X|. Note that, as we have seen above, X does have a discrete subspace of cardinality λ because λ is singular strong limit, so trivially we also have bh(X) > λ, consequently bh(X) must be a successor cardinal, sayµ⁺whereλ≤µ <|X|.

Now letG be the union of all open subsets ofX of size at mostµ. Then we have

|G| ≤µas well since otherwise we could easily produce inX a right separated subset of cardinality µ⁺ contradictingbh(X) = µ⁺. But then, in the closed subspaceX \G

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of X, clearly each non-empty relatively open set has size > µ ≥ λ. By a result of Šapirovskiˇı(see [25], 3.14) every countably tight compactum has countableπ-character, hence lemma1.11can be applied to the spaceX\Gand the cardinalλ. Consequently, we havebg(X) ≥ bg(X \G) > λ^ω ≥ |X|, and so we may again conclude thatgb(X) =

|X|⁺.

1.3 Two improvements on Tkaèenko’s addition theorem

We start this section by recalling a few well-known definitions and introducing some related notation.A spaceXis said to be aD-space if for any neighbourhood assignment φdefined onX there is a closed discrete setD ⊂Xsuch that ∪{φ(x) :x∈ D}=X.

For any spaceX we set

• D(X) = min{|A| :X=∪AandAis aD-space for eachA∈ A }.

• ls(X) = min{|A|:X =∪AandAis left-separated for eachA∈ A }.

(Note that bothD(X)and ls(X)can be finite.)

It was shown in [70] that left-separated spaces are D-spaces, hence we have D(X) ≤ ls(X)for anyX.

In [62], M. Tkaˇcenko proved the following remarkable result: IfX is a countably com- pactT₃-space withls(X)≤ωthen

(i) Xis compact, (ii) Xis scattered, (iii) Xis sequential.

It is easy to see that if in a scattered compactT₂-space any countably compact subspace is compact then it is sequential, hence (iii) immediately follows from (i) and (ii), although this is not how (iii) was proved in [62].

The aim of this section is to improve (i) and (ii) as follows:

(A) Any countably compact spaceXwithD(X)≤ωis compact.

(B) IfX is compactT₂ withls(X)< N(R)thenXis scattered.

Here N(R) denotes the Novák number of the real line R, i.e. the covering number cov(M)of the idealMof all meager subsets ofR.

If X is any crowded (i.e. dense-in-itself) space and Y ⊂ X then we denote by N(Y, X)the relative Novák number ofY inX, that is the smallest number of nowhere

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dense subsets of X needed to coverY. In particular, N(X) = N(X, X)is the Novák number ofX.

We should also mention that a weaker version of statement (A), in whichD(X)< ω is assumed instead ofD(X)≤ω, has been established in [20].

Similarly as in [62], we can actually prove the following higher-cardinal generalization of statement (A).

Theorem 1.14. Letκbe any infinite cardinal andXbe initiallyκ-compact withD(X)≤ κ. ThenX is actually compact.

The proof of Theorem 1.14 is based on the following lemma that may have some independent interest in itself.

Lemma 1.15. Let X be any space and Y ⊂ X its D subspace. If ρ is a regular cardinal such that X has no closed discrete subset of sizeρ(i.e. e(X)ˆ ≤ ρ), moreover U = {U_α : α ∈ ρ}is a strictly increasing open cover of X then there is a closed set Z ⊂X such thatZ∩Y =∅andZ 6⊂U_α for allα∈ρ.

Proof.If there is anα ∈ ρwithY ⊂ U_α thenZ = X −U_α is clearly as required. So assume from here on thatY 6⊂U_αfor allα∈ρ.

For every pointy ∈Y letα(y)be theminimalordinalαsuch thaty ∈U_αand then consider the neighbourhood assignmentφonY defined by

φ(y) = U_α(y).

SinceY is aD-space there is a setE ⊂Y, closed and discrete inY, such thatY ⊂φ[E].

We claim thatZ =E⁰, the derived set ofE, is now as required.

Indeed, Z is closed in X and Z ∩Y = ∅ as E has no limit point withinY. It remains to show thatZ 6⊂ U_αfor allα ∈ρ. Assume, indirectly, thatZ ⊂ U_αfor some α ∈ ρ. Note first that for any point y ∈ Y ∩U_α we have α(y) ≤ α, consequently φ[E ∩Uα] ⊂ Uα. On the other hand, Z = E⁰ ⊂ Uα implies that E −Uα is closed discrete inX, hence|E−U_α|< ρ by our assumption. But then

β = sup{α(y) :y∈E−U_α}< ρ becauseρis regular, consequently we have

Y ⊂φ[E] =φ[E∩Uα]∪φ[E−Uα]⊂Uα∪Uβ =Umax{α,β}, contradicting that no member ofU coversY.

Now, we can turn to the proof of our theorem.

Proof.It suffices to prove that for no regular cardinalρis there a strictly increasing open cover ofX of the form U = {U_α : α ∈ ρ}. For ρ ≤ κ this is clear, for X is initially

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κ-compact. So assume now thatρ > κ, and assume indirectly thatU = {U_α :α ∈ ρ} is a strictly increasing open cover ofX. Note also thatX has no closed discrete subset of sizeρ > κbecauseX is initiallyκ-compact.

By D(X) ≤ κ we have X = ∪{Y_ν : ν ∈ κ}, where Y_ν is a D subspace of X for eachν ∈ κ. Using lemma1.15then we may define by a straightforward transfinite recursion onν ∈κclosed setsZ_ν ⊂Xsuch that for eachν ∈ κwe haveZ_ν∩Y_ν =∅, Z_ν 6⊂U_αfor allα∈ρ, moreoverν₁ < ν₂impliesZ_ν₁ ⊃Z_ν₂. In this we make use of the fact that ifν < κand{Z_η : η ∈ ν}is a decreasing sequence of closed sets inX such that∩{Z_η :η ∈ν} ⊂ U for some openU ⊂ Xthen there is anη ∈νwithZ_η ⊂ U as well, using again the initialκ-compactness ofX.

But then, applying once more thatXis initiallyκ-compact, we conclude that

∩{Z_ν :ν ∈κ} 6=∅, contradicting thatX =∪{Y_ν :ν ∈κ}.

It should be noted that in the above result no separation axiom is needed. This is in contrast with Tkaˇcenko’s result from [62].

Let us now turn to our second statement (B). Again, we need to first give a prepara- tory result. For this we recall the cardinal functionδ(X)that was introduced in [72]:

δ(X) = sup{d(S) : S is dense in X}.

Let us note here that ifX is a compactT₂-space thenδ(X) = π(X), as was shown in [33].

Lemma 1.16. Assume thatXis an arbitrary crowded topological space andY ⊂Xis its left-separated subspace. Then we have

N(Y, X)≤δ(X), consequently

N(X)≤ls(X)·δ(X).

Proof.We shall proveN(Y, X)≤δ(X)by transfinite induction on the order type of the well-ordering that left-separatesY. So assume that≺is a left-separating well-ordering ofY such that ifZ is any proper initial segment ofY, w.r.t.≺, thenN(Z, X)≤δ(X).

Let Gbe the union of all those open setsU in X for which Y (or more precisely:

U ∩Y) is dense inU. Clearly, thenY \G is nowhere dense inX andY ∩Gis dense inG. The latter then implies

d(Y ∩G)≤δ(G)≤δ(X).

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On the other hand, since≺left-separatesY ∩G, any dense subset ofY ∩Gmust be cofinal inY ∩G w.r.t.≺, hence we clearly have

cf(Y ∩G,≺)≤d(Y ∩G)≤δ(X).

But any proper ≺-initial segment of Y ∩ Gmay be covered by δ(X) many nowhere dense sets, by the inductive hypothesis, hence we have

N(Y, X)≤1 +δ(X)·δ(X) = δ(X),

becaused(X)and soδ(X)is always infinite by definition. The second part now follows immediately.

Note that again absolutely no separation axiom was needed in the above result. How- ever, in the proof of the following theorem the assumption of Hausdorffness is essential.

Theorem 1.17. LetX be a compactT2-space satisfyingls(X) < N(R). ThenX must be scattered.

Proof.We actually prove the contrapositive form of this statement. So assume thatX is not scattered, then it is well-known that some closed subspace F ⊂ X admits an irreducible continuous closed mapf :F →Conto the Cantor setC.

It is also well-known and easy to check that then we have δ(F) = δ(C) = ω, moreoverN(F) = N(C) = N(R)> ω. But then from lemma1.16we conclude that

ls(X)≥ls(F) = ls(F)·ω≥N(F) =N(R).

We would like to mention that 1.16 and 1.17 were motivated by the treatment of Tkaˇcenko’s results given in [60]. We also point out that theorems1.14 and1.17 yield a slight strengthening of Tkaˇcenko’s theorem in that the T₃ separation axiom may be replaced by T₂ in it. This is new even in the case of left-separated spaces (i. e. the assumptionls(X) = 1) that preceded Tkaˇcenko’s result in [17].

Corollary 1.18. LetX be a countably compactT₂space that satisfiesls(X)≤ω. Then Xis compact, scattered, and sequential.

We finish by formulating a couple of natural problems concerning our results.

Problem 1.19. Is the upper bound N(R) in theorem 1.17 sharp? Can it actually be replaced by the cardinality of the continuum (in ZFC, of course)?

Note that as metric or compact spaces are allD-spaces, in theorem1.17one clearly cannot replacels(X) with D(X). Also, a compact (D-)space may fail to be sequential. Being left-separated, however, is clearly a hereditary property, hence left-separated spaces are actually hereditaryD-spaces. Thus the following problems may be raised.

Problem 1.20. Is a compactT₂hereditaryD-space sequential? Does it contain a point of countable character?

Concerning this problem we note that it follows easily from theorem 1.14 that a compactT₂-spaceXsatisfyingD(Y)≤ωfor allY ⊂X has countable tightness.

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1.4 On d-separability of powers and C

p

(X )

A space is calledd-separable if it has a dense subset representable as the union of countably many discrete subsets. Thusd-separable spaces form a common generalization of separable and metrizable spaces. A. V. Arhangelskii was the first to studyd-separable spaces in [3], where he proved for instance that any product of d-separable spaces is againd-separable. In [64], V. V. Tkachuk considered conditions under which a function space of the formC_p(X)isd-separable and also raised a number of problems concerning thed-separability of both finite and infinite powers of certain spaces. He again raised some of these problems in his lecture presented at the 2006 Prague Topology Confer- ence. In this note we give solutions to basically all his problems concerning infinite powers and to one concerningC_p(X).

Theorem 1.21. Let κ be an infinite cardinal and let X be a T₁ space satisfying bs(X^κ)> d(X).Then the powerX^κisd-separable.

Proof.IfX itself is discrete then all powers ofX are obviouslyd-separable, hence in what follows we assume thatX is not discrete. Consequently, we may pick an accumulation point ofXthat we fix from now on and denote it by0. By definition, we may then find a dense subsetS ofX with0 ∈/ Sand|S| = d(X) = δ.For any non-empty finite set of indices a ∈ [κ]^<ω we have then |Sâ| = δ as well, hence we may fix a one-one indexingSâ={sâ_ξ :ξ < δ}.

Let us next fix an increasing sequence hI_n:n < ωi of subsets of κ such that S

n<ωI_n =κand|κ\I_n|=κfor eachn < ω.It follows from our assumptions then that for everyn < ω there is adiscretesubspaceD_nof the “partial" powerX^κ\Iⁿ such that

|D_n|=δ.Thus we may also fix a one-one indexing ofD_nof the form D_n={y_ξⁿ:ξ < δ}.

The discreteness ofD_nmeans that for eachξ < δthere is an open setU_ξⁿinX^κ\Iⁿ such thatU_ξⁿ∩D_n ={yⁿ_ξ}.

Now fixn ∈ ω and pick a non-empty finite subsetaof I_n.For each ordinalξ < δ we define a pointx^{n, a}_ξ ∈X^κ as follows:

x^{n, a}_ξ (α) =







s^a_ξ(α) ifα∈a, 0 ifα∈I_n\a, yⁿ_ξ(α) ifα∈κ\I_n.

Having done this, for any n < ω and 1 ≤ k < ω we define a subset E^{n, k} ⊂ X^κ by putting

E^{n, k} ={x^{n, a}_ξ : a∈[I_n]^kandξ < δ}.

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Now, for nanda as above and forξ < δ, letW_ξ^{n, a} be the (obviously open) subset ofX^κ consisting of those pointsx ∈ X^κ that satisfy bothx(α) 6= 0for allα ∈ aand x(κ\I_n)∈U_ξⁿ.Clearly, we havex^{n, a}_ξ ∈W_ξ^{n, a} and we claim that

W_ξ^{n, a}∩E^{n, k} ={x^{n, a}_ξ }

whenever a ∈ [In]^k. Indeed, if b ∈ [In]^k anda 6= b then |a| = |b| = k implies that a\b 6=∅, hence for anyα ∈ a\b and for anyη < δ we havex^{n, b}_η (α) = 0showing that x^{n, b}_η ∈/W_ξ^{n, a}.Moreover, for any ordinalη < δwithη6=ξwe have

x^{n, a}_η (κ\I_n) = y_ηⁿ∈/ U_ξⁿ,

hence againx^{n, a}_η ∈/ W_ξ^{n, a}. Thus we have shown that each setE^{n, k} is discrete, while their union is trivially dense inX^κ. Consequently,X^κ is indeedd-separable.

Let us note now that if X is any T₁ space containing at least two points then the power X^κ includes the Cantor cube2^κ that is known to contain a discrete subspace of sizeκ.So if we apply this trivial observation toκ=d(X), then we obtain immediately from theorem 1.21 the following corollary which answers problem 4.10 of [64]. This was asking if for every (Tychonov) space X there is a cardinal κ such that X^κ is d- separable.

Corollary 1.22. For everyT₁ spaceXthe powerX^d(X)isd-separable.

Next we show that ifX is compact Hausdorff then evenX^ωisd-separable, answer- ing the second half of problem 4.2 from [64]. This will follow from the following result that we think is of independent interest.

Theorem 1.23. IfX is any compactT₂ space thenX² contains a discrete subspace of sized(X), that isbs(X²)> d(X).

Proof.Let us assume first that for every non-empty open subspaceG⊂Xwe also have w(G) ≥ d(X) = δ. We then define by transfinite induction on α < δ distinct points xα, yα ∈X together with theirdisjointopen neighbourhoodsUα, Vαas follows.

Suppose that α < δ, moreoverx_β ∈U_β andy_β ∈ V_β have already been defined for allβ < α.Thenα < δ =d(X)implies that there exists a non-empty open setG_α ⊂X such that neitherxβ noryβ belongs toGα forβ < α. Let us choose then a non-empty open setH_α such thatH_α ⊂ G_α and consider the topologyτ_α onH_α generated by the traces of the open setsU_β, V_β for allβ < α.Since

w(H_α, τ_α)< δ ≤w(H_α)≤w(H_α),

the topologyτ_αis strictly coarser than the compact Hausdorff subspace topology ofH_α inherited fromX, henceτ_α is not Hausdorff. We pick the two points x_α, y_α ∈ H_α so

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that they witness the failure of the Hausdorffness ofτ_α. Note that, in particular, this will imply

hx_α, y_αi∈/U_β ×V_β

for allβ < α.We may then choose their disjoint open (inX) neighbourhoodsU_α, V_α insideG_α. This will clearly imply that we shall also havehx_α, y_αi∈/ U_γ×V_γwhenever α < γ < δ.Thus, indeed,{hx_α, y_αi:α < δ}is a discrete subspace ofX².

Now, assume that X is an arbitrary compact Hausdorff space and call an open set G ⊂ X goodif we have d(H) = d(G) for every non-empty openH ⊂ G. Clearly, every non-empty open set has a non-empty good open subset, hence ifG is a maximal disjoint family of good open sets inX thenS

Gis dense inX. Consequently we have X{d(G) :G∈ G} ≥d(X).

But for every G ∈ G its squareG² has a discrete subspaceD_G with|D_G| = d(G).

Indeed, ifHis open with∅ 6=H ⊂Gthen for every non-empty openU ⊂H we have w(U) ≥ d(U) = d(H) = d(H), so the first part of our proof applies toH, that isH² (and thereforeG²) has a discrete subspace of sized(H) =d(U).It immediately follows thatD=S

{D_G :G∈ G}is discrete inX², moreover

|D|=X

{d(G) :G∈ G} ≥d(X), completing our proof.

Any compact L-space, more precisely: a non-separable hereditarily Lindelof compact space (e. g. a Suslin line), demonstrates, alas only consistently, that in theorem 1.23the squareX²cannot be replaced byXitself. On the other hand, we should recall here Shapirovskii’s celebrated result, see 3.13 of [25], which states thatd(X)≤s(X)⁺ holds for any compactT2 spaceX. This leads us to the following natural question.

Problem 1.24. Is there a ZFC example of a compactT₂ spaceXthat does not contain a discrete subspace of cardinalityd(X)?

SinceX²embeds as a subspace intoX^ω, theorems1.21and1.23immediately imply the following.

Corollary 1.25. IfXis any compactT₂ space thenX^ωisd-separable.

Of course, to get corollary 1.25 it would suffice to know bs(X^ω) > d(X). Our next result shows, however, that if we know that some finite power ofX has a discrete subspace of sized(X)then we may actually obtain a stronger conclusion. To formulate this result we again fix a point0∈X and introduce the notation

σ(X^ω) =

x∈X^ω : {i < ω :x(i)6= 0}is finite .

Clearly, σ(X^ω)is dense in X^ω, hence the d-separability of the former implies that of the latter.

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Theorem 1.26. Let X be a space such that, for some k < ω, the power X^k has a discrete subspace of cardinality d(X).Thenσ(X^ω)(and henceX^ω) isd-separable.

Proof. Let us put again d(X) = δ and fix a dense set S ⊂ X with |S| = δ. By assumption, there is a discrete subspace D ⊂ X^k with a one-one indexingD = {d_ξ : ξ < δ}.Also, for each natural numbern ≥1we have|Sⁿ|=δ, so we may fix a one-one indexingSⁿ={sⁿ_ξ :ξ < δ}.

Now, for any1≤n < ωandξ < δwe define a pointxⁿ_ξ ∈σ(X^ω)with the following stipulations:

xⁿ_ξ(i) =







sⁿ_ξ(i) ifi < n,

d_ξ(i−n) ifn≤i < n+k, 0 ifn+k≤i < ω.

It is straight-forward to check that eachD_n = {xⁿ_ξ : ξ < δ} ⊂ σ(X^ω)is discrete, moreoverS

n<ωDnis dense inσ(X^ω).

Actually, before we get too excited, let us point out that the d-separability of X^ω implies that some finite power of X has a discrete subspace of cardinality d(X), in

“most" cases, namely ifcf(d(X))> ω.Indeed, first of all, in this case there is a discrete D ⊂ X^ω with |D| = d(X^ω) = d(X). Secondly, for each point x ∈ D there is a finite set of co-ordinates a_x ∈ [ω]^<ω that supports a neighbourhoodU_x ofx such that D∩U_x ={x}.But bycf(|D|)> ωthen there is somea∈[ω]^<ωwith|{x∈D :a_x = a}|=|D|=d(X), and we are clearly done.

Let us mention though that thed-separability of the powerX^ωdoes not imply that of some finite power ofX.In fact, the ˇCech–Stone remainderω^∗demonstrates this because itsω^thpower isd-separable by theorem1.26but no finite power ofω^∗isd-separable, as it was pointed out in [64, 3.16 (b)].

Next we give a negative solution to one more problem of Tkachuk concerning thed- separability of powers. Problem 4.9 from [64] asks if thed-separability of some infinite powerX^κimplies thed-separability of the countable powerX^ω.We recall that a strong L-space is a non-separable regular space all finite powers of which are hereditarily Lin- delöf.

Theorem 1.27. LetX be a strong L-space with d(X) = ω₁.ThenX^ω¹ isd-separable butX^ω is not. Moreover, there is a ZFC example of a0-dimensionalT₂ spaceY such thatY^ω² isd-separable butY^ω¹ (and henceY^ω) is not.

Proof.It is immediate from corollary1.22thatX^ω¹ isd-separable. Also, since all finite powers ofX are hereditarily Lindelöf so isX^ω, hence

s(X^ω) =ω < ω1 =d(X^ω) implies thatX^ωcannot bed-separable.

On the structure of compact topological spaces