A spaceX isC-Lindel¨of (weaklyC-Lindel¨of) if for every closed subsetF ofX and every open coverUofF by open subsets ofX, there exists a countable subfamilyVofUsuch thatF ⊆ ∪{V :V ∈ V}(respectively, F ⊆ ∪V)

60 (2008), 173–180 originalni nauqni rad research paper

SOME TOPOLOGICAL PROPERTIES WEAKER THAN LINDEL ¨OFNESS

Yan-Kui Song

Abstract. A spaceX isC-Lindel¨of (weaklyC-Lindel¨of) if for every closed subsetF ofX and every open coverUofF by open subsets ofX, there exists a countable subfamilyVofUsuch thatF ⊆ ∪{V :V ∈ V}(respectively, F ⊆ ∪V). In this paper, we investigate the relationships amongC-Lindel¨of spaces, weaklyC-Lindel¨of spaces and Lindel¨of spaces, and also study various properties of weaklyC-Lindel¨of spaces andC-Lindel¨of spaces.

1. Introduction

By a space, we mean a topological space. In 1969, Viglino [2] introduced the concept of C-compact spaces that is weaker than compactness. Recall that a space X is C-compact if for every closed subset F of X and every open cover U of F by open subsets of X, there exists a finite subfamily V of U such that F ⊆S

{V :V ∈ V}. It is well-known that a spaceX is Lindel¨ofif for every open cover of X has a countable subcover. As motivations of the classes ofC-compact spaces and Lindel¨of spaces, we give the following classes of spaces:

Definition 1.1. A space X is C-Lindel¨of if for every closed subset F of X and every open cover U of F by open subsets of X, there exists a countable subfamilyV ofU such thatF ⊆S

{V :V ∈ V}.

Definition 1.2. A spaceX isweaklyC-Lindel¨of if for every closed subset F ofX and every open cover U ofF by open subsets ofX, there exists a countable subfamilyV ofU such thatF ⊆ ∪V.

From the above definitions, it is clear that ifXis Lindel¨of, thenXisC-Lindel¨of and if X is C-Lindel¨of, then X is weakly C-Lindel¨of. But, the converses do not hold in the class of Hausdorff spaces or the class of Tychonoff spaces (see below Examples 2.3 and 2.4).

AMS Subject Classification: 54D15, 54D20.

Keywords and phrases: Lindel¨of,C-Lindel¨of, weaklyC-Lindel¨of.

The author acknowledges the support from the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (Grant No 07KJB110055)and NSFC Projects 10571081

173

The purpose of this paper is to investigate the relationship betweenC-Lindel¨of spaces, weaklyC-Lindel¨of spaces and Lindel¨of spaces, and also study various prop- erties of weaklyC-Lindel¨of spaces andC-Lindel¨of spaces.

Throughout this paper, the cardinality of a set A is denoted by |A|. Let ω denote the first infinite cardinal,ℵ1the first uncountable cardinal,cthe cardinality of the continuum. Other terms and symbols that we do not define will be used as in [1].

2. Some examples onC-Lindel¨of spaces and weakly C-Lindel¨of spaces

In this section, we clarify the relationships of these spaces given in the first section by giving some examples. First, the following theorem can be easily proved:

Theorem 2.1. If X is a regularC-Lindel¨of space, then every closed subset of X is Lindel¨of.

Corollary 2.2. If X is a regular C-Lindel¨of space, then X is Lindel¨of.

In the following, we give an example showing that Corollary 2.2 does not hold for the class of Hausdorff spaces.

Example 2.3. There exists a Hausdorff C-Lindel¨of space X which is not Lindel¨of.

Proof. Let

A={aα:α <ℵ1}, B ={bβ:β <ℵ1}andY ={haα, bβi:α <ℵ1, β <ℵ1} Since |B| =ℵ1, we can write B as B =∪α<ℵ1Bα such that |Bα| = ℵ1 for each α <ℵ1andBα∩Bα⁰ =∅forα⁰6=α. For eachα <ℵ1, letAα={haα, bβi:β <ℵ1}.

Let

X =Y ∪A∪ {a} wherea /∈Y ∪A.

We topologize X as follows: every point ofY is isolated; a basic neighborhood of a pointaα∈Afor eachα <ℵ1takes the from

Uaα(γ) ={aα} ∪ {haα, bβi:β > γ} ∪ {haδ, bβi:bβ ∈Bα, δ > γ}forγ <ℵ1

and a basic neighborhood ofatakes the from Ua(α) ={a} ∪ [

β>α

{haγ, bδi:bδ∈Bβ, γ > α}forα <ℵ1.

Clearly,Xis a Hausdorff space by the construction of the topology onX. Moreover, X is not regular, since the pointacannot be separated from the closed subsetAby disjoint open subsets ofX. SinceAis a discrete closed subset ofX with|A|=ℵ1, thenX is not Lindel¨of.

Let us show that X is C-Lindel¨of. LetF be a closed subset of X and U an open cover of F by open subsets ofX. Without loss of generality, we can assume thatU consists of basic open sets ofX.

Case (1): a∈F.

Sincea∈F, there is aUa∈ U such that a∈Ua. By assumption, there exists aα0<ℵ1such that

Ua =Ua(α0) ={a} ∪ ∪β>α0{haγ, bδi:bδ ∈Bβ, γ > α0}.

By definition of the topology ofX, we have

F∩({aβ:β > α0} ∪Ua(α0))⊆Ua(α0).

Let A0 ={α: aα ∈F ∩ {aβ :β < α0+ 1}} and A1 ={α: aα ∈/ F∩ {aβ : β <

α0+ 1}}. ThenA0 andA1 are countable.

Forα∈A0,aα∈F and there is aUaα∈ Usuch thataα∈Uaα. By assumption, there is aαγ <ℵ1 such that

Uaα =Uaα(αγ) ={aα} ∪ {haα, bβi:β > αγ} ∪ {haδ, bβi:bβ ∈Bα andδ > αγ}.

Forα∈A0, sinceF∩ {haα, bβi:β < αγ+ 1}is at most countable, there exists a countable subfamilyVα ofU such that

F∩ {haα, bβi:β < αγ+ 1} ⊆ ∪{V :V ∈ Vα}.

LetUα={Uaα} ∪ Vα. ThenUαis a countable subfamily of U and F∩(Uaα(αγ)∪ {haα, bβi:β < αγ+ 1})⊆ ∪{U :U ∈ Uα}.

If we putU⁰ =S

α∈A0Uα,U⁰ is a countably subfamily ofU and [

α∈A0

(F∩(Uaα(αγ)∪ {haα, bβi:β < αγ+ 1}))⊆[

{U :V ∈ U⁰}.

On the other hand, forα∈A1,aα∈/F, sinceF is closed, there exists an open neighborhoodUaα(αγ) ofaαfor someαγ <ℵ1such that

Uaα(αγ)∩F =∅.

Therefore,F∩{haα, bβi:β <ℵ1}is at most countable, and there exists a countable subfamilyVα ofU such that

F∩Aα⊆ ∪{U :U ∈ Vα}.

If we putU⁰⁰=S

α∈A1V_α, thenU⁰⁰is a countably subfamily ofU and [

α∈A1

(F∩Aα)⊆[

{U :U ∈ U⁰⁰}

Letα⁰ = sup{αγ :α∈A0∪A1}. Thenα⁰<ℵ1, sinceA0∪A1 is countable. If we putU0=U⁰∪ U⁰⁰, then

F∩( [

α<α0+1

({aα} ∪Aα∪ {haδ, bβi:bβ ∈Bα, δ > α⁰}))⊆ ∪{U :U ∈ U0}.

For eachα0 < α < α⁰+ 1, it is not difficult to find a countable subfamilyUαofU such that

F∩({aα} ∪Aα)⊆[

{U :U ∈ Uα}.

LetU1=S

α0<α<α⁰+1Uα. ThenU1 is countable subfamily ofU and F∩( [

α0<α<α⁰+1

({aα} ∪Aα))⊆[

{U :U ∈ U1}.

If we put V = {Uaα} ∪ U0∪ U1, then V is a countable subfamily of U and F ⊆ S{U :U ∈ V}, which completes the proof.

Case (2): a /∈F.

Sincea /∈F, there is a basic open neighborhoodUa ofasuch thatUa∩F =∅.

Without loss of generality, we can assume that Ua=Ua(α0) ={a} ∪ [

β>α0

{haγ, bδi:bδ ∈Bβ, γ > α0}for someα0<ℵ1.

As in the previous case, we can find aα⁰ <ℵ1 and a countable subfamilyU0 ofU such that

F∩( [

α<α0+1

({aα} ∪Aα∪ {haδ, bβi:bβ∈Bα, δ > α⁰} ⊆[

{U :U ∈ U0}.

IfF∩ {aα:α > α0}=∅, similarly as in the proof above, we can find a countable subfamilyU1ofU such that

F∩( [

α0<α<α⁰+1

({aα} ∪Aα))⊆[

{U :U ∈ U1}.

If we putV=U0∪U1, thenVis a countable subfamily ofU andF ⊆ ∪{U :U ∈ V}.

On the other hand; ifF∩ {aα:α > α0} 6=∅, we can pickaβ0 ∈F∩ {aα:α >

α0}, and there isU ∈ U such thataβ0 ∈U, and we can assume

U =Uaβ0(γ) ={aβ0} ∪ {haβ0, bβi:β≥γ} ∪ {haδ, bβi:bβ∈Bβ0, δ > γ}forγ <ℵ1. Then

F∩ {aα:α > γ} ⊆U .

Forα0< α <max{α⁰, γ+ 1}+ 1 =γ⁰, we can find a countable subfamilyUαofU such that

F∩({a } ∪A )⊆[

{U :U ∈ U }.

If we putU1=S

α0<α<γ⁰Uα, then F∩( [

α0<α<γ⁰

({aα} ∪Aα))⊆[

{U :U ∈ U1}

If we putV ={U} ∪ U0∪ U1, thenV is a countable subfamily ofU andC⊆ ∪{U : U ∈ V},which completes the proof.

Example 2.4. There exists a Tychonoff weakly C-Lindel¨of space X that is notC-Lindel¨of.

Proof. LetX =ω∪ Rbe the well-known Mr´owka space, whereRis a maximal almost disjoint family of infinite subsets ofωwith|R|=c(see [3]).

We show that X is not C-Lindel¨of. Since|R| = c, we can enumerate R as {rα:α <c}. LetF={rα:α <c}. ThenF is a closed subset ofX.

Let

Uα={rα} ∪rαfor eachα <c.

ThenUα is a closed and open subset ofX. Let us consider the open cover U ={Uα:α <c}

ofF. For any countable subfamilyV ofU, letα0= sup{α:Uα∈ V}. Thenα0<c, sinceV is countable. If we pickα⁰ > α0, thenrα⁰ ∈ ∪{U/ :U ∈ V}, since Uα⁰ ∈ V/ and Uα⁰ is the only element of U containing rα⁰ and Uα⁰ ∩Uα is finite for each α < α0, which shows thatX is notC-Lindel¨of.

Next, we show thatX is weakly C-Lindel¨of. Let F be any closed subset of X and U any open cover of F by open subsets of X. Without loss of generality, we assume that U consists of basic open sets of X. Let A = F∩ {rα : α < c}. For eachrα∈Athere is aVα∈ U such that rα∈Vα. By assumption, there is a finite subsetFα ofω such that

Vα={rα} ∪(rα\Fα).

LetC=∪{rα\Fα:rα∈A}. ThenC is a countable subset ofω. For eachn∈C we pickVn∈ U such that n∈Vn. LetV1={Vn :n∈C}. Then V1 is a countable subfamily ofU. By the construction of the Mr´owka space, it is not difficult to show that

A⊆ ∪V1.

LetB =F ∩ω. ThenB is a countable subset of ω, since ω is countable. Hence, there exists a countable subfamilyV2 ofU such that

B⊆ ∪V2.

If we put V =V1∪ V2, then V is a countable subfamily of U such that F ⊆ ∪V, which shows thatX is weaklyC-Lindel¨of.

3. Various properties of weakly C-Lindel¨of spaces andC-Lindel¨of spaces From Example 2.4, it is not difficult to see that the closed subset Rof X is not weakly C-Lindel¨of, which shows that a closed subset of a weakly C-Lindel¨of space need not be weaklyC-Lindel¨of. In the following, we give a stronger example that shows that a regular closed subspace of a Tychonoff weakly C-Lindel¨of space need not be weaklyC-Lindel¨of.

Example 3.1. There exists a Tychonoff weaklyC-Lindel¨of space X having a regular closed subspace which is not weakly C-Lindel¨of.

Proof. Let S1 = ω∪ R be the same Isbell-Mr´owka space as in the proof of Example 2.4. ThenS1 is weaklyC-Lindel¨of.

LetDbe a discrete space of cardinality c, and let S2= (βD×(ω+ 1))\((βD\D)× {ω}) be the subspace of the product ofβDandω+ 1.

We show thatS2 is not weaklyC-Lindel¨of. Since|D|=c, we can enumerate D as{dα:α <c}. LetF ={hdα, ωi:α <c}. ThenF is a closed subset ofX.

Let

Uα={dα} ×[0, ω] for eachα <c.

ThenUα is a closed and open subset ofS2. Let us consider the open cover U ={Uα:α <c}

ofF. For any countable subfamilyV ofU letα0= sup{α:Uα∈ V}. Thenα0<c, sinceV is countable. If we pickα⁰ > α0, thenhdα⁰, ωi∈ ∪V, since/ Uα⁰ is the only element ofU containinghdα⁰, ωiand Uα⁰∩Uα =∅ for eachα < α0, which shows thatS2is not weaklyC-Lindel¨of.

We assume thatS₁∩S₂=∅. Since|R|=c, we can enumerateRas{r_α:α <

c}. Letϕ:D× {ω} → R be a bijection by

ϕ(hdα, ωi) =rα for eachα <c.

LetX be the quotient space obtained from the discrete sumS1⊕S2 by identifying hdα, ωiwith rα for each α < c. Let π: S1⊕S2 → X be the quotient map. Let Y =π(S2). Then Y is a regular closed subspace ofX, however, it is not weakly C-Lindel¨of, since it is homeomorphic toS2.

Now, we show thatX is weaklyC-Lindel¨of. For that purpose, letF be a closed subset ofX andU an open cover ofF by open subsets ofX. Let

F⁰ =F∩π(S1) andFn=F∩π(βD× {n}) for each n∈ω.

Since S1 is weakly C-Lindel¨of, π(S1) is weakly C-Lindel¨of, hence there exists a countable subfamily U⁰ of U such that F⁰ ⊆ S

U⁰. For each n ∈ ω, since Fn

is compact, there exists a finite subfamily U such that F ⊆ S

U . If we put

V =U⁰∪ {Un:n∈ω}, then V is a countable subfamily ofU andF ⊆S

V, which shows thatX is weaklyC-Lindel¨of.

The following theorem can be easily proved.

Theorem 3.2. IfA is a closed and open subset of a weaklyC-Lindel¨of spaces X, thenAis weaklyC-Lindel¨of.

For aC-Lindel¨of space, by Corollary 2.2, every closed subset of a regular C- Lindel¨of space is Lindel¨of (hence,C-Lindel¨of). From Example 2.3, it is not difficult to see that the closed subset{aα:α <ℵ1}of a Hausdorff spaceXis notC-Lindel¨of which shows that a closed subset of a HausdorffC-Lindel¨of space need not be C- Lindel¨of. In the following, we give a stronger example that shows that a regular closed subspace of a HausdorffC-Lindel¨of space need not beC-Lindel¨of.

Example 3.3. There exists a HausdorffC-Lindel¨of space X having a regular closed subspace which is not weaklyC-Lindel¨of.

Proof. LetS1=X be the same spaceX as in the proof of Example 2.3. Then S1 is a HaudorffC-Lindel¨of space.

LetDbe a discrete space of cardinality ℵ1, and let S2= (βD×(ω+ 1))\((βD\D)× {ω})

be the subspace of the product of βD andω+ 1. Similar to the proof that S2 is not weaklyC-Lindel¨of in Example 3.1, we can prove thatS2is notC-Lindel¨of.

We assume thatS1∩S2=∅. Since|D|=ℵ1, we can enumerateDas{dα<ℵ1}.

Letϕ:D× {ω} →Abe a bijection by

ϕ(hdα, ωi) =aαfor eachα <ℵ1.

LetX be the quotient space obtained from the discrete sumS1⊕S2 by identifying hdα, ωiwith aα for each α <ℵ1. Letπ:S1⊕S2 →X be the quotient map. Let Y =π(S2). ThenY is a regular closed subspace ofX, however it is notC-Lindel¨of, since it is homeomorphic toS2. Similar to the proof that X is weakly C-Lindel¨of in Example 2.3, it is not difficult to show that X is C-Lindel¨of, which completes the proof.

Now the following theorem can be easily proved.

Theorem 3.4. If A is a closed and open subset of a C-Lindel¨of spaces X, thenA isC-Lindel¨of.

Next, we consider the images of C-Lindel¨of spaces and weakly C-Lindel¨of spaces under continuous mapping. Since a continuous image of a Lindel¨of space is Lindel¨of, we give two parallel results forC-Lindel¨of spaces and weaklyC-Lindel¨of spaces.

Theorem 3.5. Let f: X → Y be a continuous mapping from a C-Lindel¨of spaceX onto a spaceY. ThenY isC-Lindel¨of.

Proof. LetF be a closed subset ofY and{Uα:α∈Λ}an open cover ofF by open subsets of Y. Then {f⁻¹(Uα) : α∈Λ} is an open cover off⁻¹(F) by open subsets ofX. SinceX isC-Lindel¨of, there exists a countable subset{αi:i∈ω}of Λ such thatf⁻¹(F)⊆S

i∈ωf⁻¹(Uαi) and thus F =f(f⁻¹(F))⊆[

i∈ω

f(f⁻¹(Uαi))⊆ [

i∈ω

f(f⁻¹(Uαi)))⊆ [

i∈ω

Uαi. Hence,Y isC-Lindel¨of, which completes the proof.

Similar to the proof of Theorem 3.5, we can prove the following theorem.

Theorem 3.6. Let f: X → Y be a continuous mapping from a weakly C- Lindel¨of spaceX onto a spaceY. ThenY is weaklyC-Lindel¨of.

Now, we turn to consider preimages. To show that the preimage of a weakly C-Lindel¨of space under a closed 2-to-1 continuous map need not be weakly C- Lindel¨of, we use the Alexandorff duplicateA(X) of a spaceX. The underlying set ofA(X) isX× {0,1}; each point ofX× {1} is isolated and a basic neighborhood of a pointhx,0i ∈X× {0} is a set of the form (U × {0})∪(U × {1})\ {hx,1i}), whereU is a neighborhood ofxinX.

Example 3.7. There exists a 2-to-1 closed continuous mapf from a space X to a weaklyC-Lindel¨of spaceY such thatX is not weaklyC-Lindel¨of.

Proof. LetY be the same spaceX as in the proof Example 2.4 and consider the spaceX =A(Y). Letf:X →Y be the projection. Then f is a 2-to-1 closed continuous map. The space Y is weaklyC-Lindel¨of by Example 2.4, but X is not weakly C-Lindel¨of, sinceR × {1} is a discrete open and closed subset of X with

|R × {1}|=c.

By considering the Alexandroff duplicate of the space Y in Example 2.3, in the same manner we can prove that the preimage of a C-Lindel¨of space under a closed 2-to-1 continuous map need not beC-Lindel¨of.

Remark. The author does not know if the product of twoC-Lindel¨of spaces is C-Lindel¨of and the product of two weaklyC-Lindel¨of spaces is weaklyC-Lindel¨of even if the product of aC-Lindel¨of space and a compact space, and the product of a weaklyC-Lindel¨of space and a compact space.

Acknowledgments. The author is most grateful to the referees for their kind help and valuable suggestions.

REFERENCES

[1] R. Engelking,General Topology, Revised and completed edition, Heldermann Verlag, Berlin, 1989.

[2] G. Viglino,C-compact spaces, Duke J. Math36(1969), 761–764.

[3] S. Mr´owka,Set-theoretic constructions in topology, Fund. Math.44, 2 (1977), 83–92.

(received 17.07.2007, in revised form 13.03.2008)

Department of Mathematics, Nanjing Normal University, Nanjing, 210097 P.R. China E-mail:[email protected]