Compact and extremally disconnected spaces

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PII. S0161171204208249 http://ijmms.hindawi.com © Hindawi Publishing Corp.

COMPACT AND EXTREMALLY DISCONNECTED SPACES

BHAMINI M. P. NAYAR

Received 22 August 2002

To Prof. James E. Joseph on His 65th Birthday

Viglino defined a Hausdorff topological space to beC-compact if each closed subset of the space is anH-set in the sense of Veliˇcko. In this paper, we study the class of Hausdorff spaces characterized by the property that each closed subset is anS-set in the sense of Dickman and Krystock. Such spaces are calledC-s-compact. Recently, the notion of strongly subclosed relation, introduced by Joseph, has been utilized to characterizeC-compact spaces as those with the property that each function from the space to a Hausdorff space with a strongly subclosed inverse is closed. Here, it is shown thatC-s-compact spaces are characterized by the property that each function from the space to a Hausdorff space with a strongly sub-semiclosed inverse is a closed function. It is established that this class of spaces is the same as the class of Hausdorff, compact, and extremally disconnected spaces. The class of C-s-compact spaces is properly contained in the class ofC-compact spaces as well as in the class ofS-closed spaces of Thompson. In general, a compact space need not beC-s-compact. The product of twoC-s-compact spaces need not beC-s-compact.

2000 Mathematics Subject Classiﬁcation: 54D25, 54C99, 54D30, 54G05.

1. Introduction. It is well known that every continuous function from a compact space to a Hausdorff space is a closed function. In 1969, Viglino [15] introduced the family ofC-compact spaces, showing that every continuous function from aC-compact space to a Hausdorff space is a closed function, and that this class of spaces properly contains the class of compact spaces. A spaceXisC-compactif for every closed subset AofXand for every cover ofAby open subsets ofX, there is a finite subfamily whose closures inX will coverA. In 1970, Goss and Viglino [4] provided a Hausdorff space which is notC-compact but which has the property that every continuous function from the space to a Hausdorff space is a closed function. In 1969, Dickman and Zame [3] introduced functionally compact spaces, a class of spaces which is characterized by the property that every continuous function from any one of its members to a Hausdorff space is a closed function. Since then, a number of researchers (see [1,4,5,8,10,12] to mention a few) have studied these classes of spaces and a great deal of effort has been given to the problem of identifying a class of functions such that a spaceXis C-compact if and only if each function in this class from the spaceXto a Hausdorff space is closed. Very often, the result of such an effort was the introduction of new classes of spaces. Throughout this paper, all spaces are assumed to be Hausdorff. Definitions of the concepts used are given inSection 2.

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Theorem1.1. A spaceXisC-compact if and only if all functions onXwith strongly

subclosed inverses are closed functions.

Motivated by this discovery, in [8], Joseph et al. studied the following two classes of spaces, each of which contains the class of countably compact spaces as well as the class ofC-compact spaces.

(1) Each continuous function from the space maps closed subsets onto sequentially closed subsets.

(2) Each function on the space with a strongly subclosed inverse maps closed subsets onto sequentially closed subsets.

Spaces satisfying (1) are called sequentially functionally compact and those satisfying (2) are called sequentiallyC-compact. Spaces satisfying (2) also satisfy (1). The present paper is also a study in this direction. In 1976, Thompson [13] introduced the concept of anS-closed space while studying compact extremally disconnected Hausdorff spaces in anH-closed model using the concept of a semiopen subset of Levine [9]. A spaceX isS-closedif every cover ofXwith semiopen sets has a finite subfamily whose closures coverX. This concept generated considerable interest among researchers in this field. Parallel to the concept of anH-set, the concept of anS-set was introduced by Dickman and Krystock [2]. A subsetAofX is anS-set if every cover ofAby semiopen subsets ofXhas a finite subfamily whose closures inXcoverA. In this paper, a spaceXis C-s-compact if every cover of every closed subsetAofX, by semiopen subsets ofX, has a finite subfamily whose closures inXcoverA. In other words, aC-s-compact space is a space for which every closed subset is anS-set. Here, using the concept of functions with strongly sub-semiclosed graphs introduced by Joseph and Kwack [7], the class of C-s-compact spaces is characterized by each of the following:

(a) each function from X to a Hausdorﬀ space Y with a strongly sub-semiclosed inverse is a closed function;

(b) each function from X to a Hausdorﬀ space Y with a strongly sub-semiclosed inverse is a semiclosed function.

In [1], Clay and Joseph proved that a Hausdorﬀ spaceY isC-compact if and only if every strongly subclosed relation from a Hausdorﬀ spaceXtoY is upper semicon-tinuous, denoted as u.s.c. Recall that a relation R fromX toY is a subset of X×Y, where the domain ofRisX. A relationRfromXtoY will be denoted byR⊆X×Y. In Theorem 3.10, we show thata spaceY isC-s-compact if and only if for every spaceX, every strongly sub-semiclosed relation fromXtoY is u.s.c.

In [3], Dickman and Zame defined a space to befunctionally compact if each open filterbaseΩon the space satisfying adΩ=_ΩFconverges to_ΩF.Theorem 3.2 estab-lishes thata spaceXisC-s-compact if and only if each open filterbaseΩis an open-set base forsadΩ.

In 1966, Veliˇcko [14] introduced the concept of anH-set. A subsetAof a Hausdorﬀ spaceXis anH-setif every cover ofA, by open subsets ofX, contains a ﬁnite subfamily whose closures inXcoverA. Since the introduction of this concept, it has been widely used in the study of H-closed-type properties. In fact, C-compact spaces are those spaces for which every closed subset is anH-set.

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Theorem1.2. A spaceXisC-s-compact if and only if every nonempty proper closed

subset ofXis anS-set.

In view of the above result and [2, Proposition 4.1], in the class of Hausdorff spaces, the property of beingC-s-compact is equivalent to the property of being compact and extremally disconnected. It is known that every compact and extremally disconnected space is homeomorphic to the Stone space of some complete Boolean algebra. Thus aC-s-compact space is homeomorphic to the Stone space of some complete Boolean algebra. InSection 3, characterizations of and related results forC-s-compact spaces are obtained using covers, filterbases, relations satisfying certain graph conditions, and S-sets. InSection 4, examples are given to distinguish the class ofC-s-compact spaces from the classes of compact spaces,S-closed spaces, andC-compact spaces. It is clear that aC-s-compact space isC-compact as well asS-closed. Examples are given to show that anS-closed space need not beC-s-compact. Also, aC-compact space might fail to beC-s-compact. An example of a compact space which is notC-s-compact is provided. Products of twoC-s-compact spaces need not beC-s-compact. The basic definitions and notations used in the paper are found in the following section.

2. Preliminaries. LetAbe a subset of a spaceX. Closure and interior ofAare de-noted by clAand intA, respectively. The family of all open subsets ofXcontainingA will be denoted byΣ(A) (Σ(x),ifA= {x}). A setA⊆Xis said to be semiopen[9] if there is an open setU⊆Xsuch thatU⊆A⊆clU. The family of all semiopen subsets ofXcontainingAis denoted by SO(A)(SO(x), ifA= {x}). A pointxis asemi-interior pointofAif there is aB∈SO(x)contained inA. The semi-interior ofAis denoted by sintA, andAis semiopen if and only ifA=sintA. A pointxis in the semiclosure ofAif V∩A= ∅for everyV∈SO(x). The semiclosure ofAis denoted by sclA, andAis semi-closed ifA=sclA. Theθ-closure[14] (resp.,θ-semiclosure[7]) ofA, denoted as clθA, (resp., sclθA) isΣ(A)clV(resp.,{x∈Xsuch that clV∩A= ∅for everyV∈SO(x)}). A setA⊆Xisregularly open(resp.,regularly closed) ifA=int(clA)(resp.,A=cl(intA)) or equivalently interior of a closed set (resp., closure of an open set). Since the family of regularly closed subsets of a spaceXis the same as the family of closures of semiopen subsets ofX, thenx∈sclθF if and only if for every regularly closed setV ⊆X con-taining x, V∩F = ∅. Theθ-adherence [14] (resp., s-adherence, s-θ-adherence) of a ﬁlterbaseΩ, denoted as adθΩ(resp., sadΩ, sadθΩ) on X, isΩclθF (resp.,ΩsclF,

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a strongly subclosed relation R⊆X×Y is strongly sub-semiclosed. It is known that continuous, and indeedθ-continuous, functions have strongly subclosed inverses. A relationR ⊆X×Y is u.s.c. [1], if for eachx∈X and W ∈Σ(R(x)) inY, there is a V∈Σ(x)inXsuch thatR(V )⊆W. A functionf:X→Y is said to besemiclosed[11] iff (A)is semiclosed for each closedA⊆X. A spaceX isextremally disconnected if the closure of every open set is open. Therefore, in an extremally disconnected space, every regularly closed set is open. Throughout the paper, unless otherwise stated, all operations are taken with respect to the original space.

3. C-s-compact spaces

Definition_3.1. _{A space}(X,T )_is C-s-compact _{if, whenever}A _{is a closed subset}

ofX andΩis aT-semiopen cover ofA, there is a ﬁnite subfamilyΛofΩsuch that A⊆ΛclU, closures taken in the original topology.

Equivalently,XisC-s-compact if every closed subset isS-closed, relative toX. Ob-viously, aC-s-compact space isC-compact as well asS-closed. But as will be seen in Section 4, anS-closed space as well as aC-compact space might fail to beC-s-compact.

Theorem_3.2. _{The following are equivalent for a space}(X,T ):

(a) XisC-s-compact;

(b) for each closed setA⊆Xand each filterbaseΩonA,A∩sadθΩ= ∅; (c) for each closed setA⊆Xand each open filterbaseΩonA,A∩sadΩ= ∅; (d) each open filterbase onXiss-adherent convergent;

(e) each open ﬁlterbaseΩis an open-set base forsadΩ;

(f) ifAis a closed subset ofXandΓ is a family of semiclosed subsets ofXsuch that (_ΓG)∩A= ∅, then there is a ﬁniteΛ⊆Γ such that(_ΛintG)∩A= ∅;

(g) ifAis a closed subset ofXandΩis an open ﬁlterbase onXwhose members have nonempty intersection withA, then there is ans-adherent point ofΩinA; (h) given a closed subsetAofX, aT-semiopen coverΓ ofX−A, and an open

neigh-borhoodVofA, there is a ﬁniteΛ⊆Γ such thatX=V∪(_ΛclU).

Proof. The proof will follow the sequence (a)⇒(b)⇒(c)⇒(d)⇒(e)⇒(f)⇒(g)⇒(a), (g)⇒

(h)⇒(d).

(a)⇒(b). LetXbeC-s-compact and letAbe a closed subset ofX. LetΩbe a ﬁlterbase onAand suppose thatA∩sadθΩ= ∅. Then for eacha∈A, there is anFa∈Ωand a∈sclθFa. Hence, for eacha∈A, there is aUa∈SO(a)such that clUa∩Fa= ∅. The familyᐁ= {Ua :a∈A}is a semiopen cover of A. Choose a ﬁnite B⊆A such that A⊆BclUa. Now, chooseF∈Ωsuch thatF ⊆BFa. A contradiction of the fact that clUa∩Fa= ∅for everya∈Bis reached since clUa∩F= ∅for somea∈BandF⊆Fa for everya∈B.

(b)⇒(c). LetAbe a closed subset ofXand letΩbe an open ﬁlterbase onA. Suppose thatA∩sad(Ω)= ∅. Then for eacha∈A, there is anFa∈Ωand aUa∈SO(a)such thatUa∩Fa= ∅. SinceFais open, it follows that for eacha∈A, clUa∩Fa= ∅. This contradicts (b).

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cover of the closed setX−U. Hence, there is a ﬁniteΛ⊆Ωsuch thatX−U⊆Λcl(X− sclF)⊆Λ(X−F)=X−ΛF. Hence,ΛF⊆U. Thus,Ωiss-adherent convergent.

(d)⇒(e).The proof follows immediately.

(e)⇒(f). LetA be a closed subset ofX and letΓ be a family of semiclosed subsets ofXsuch that(_ΓU)∩A= ∅. Suppose that for every finiteΛ⊆Γ,(_ΛintU)∩A= ∅. Then{(_ΛintU)∩A:Λfinite,Λ⊆Γ}is an open filterbase onA. Hence,ᐂ= {(_ΛintU):

Λﬁnite,Λ⊆Γ}is an open ﬁlterbase onX. Then

sadᐂ=B=

ᐂ

sclV=

ﬁniteΛ⊆Γ scl

Λ intU

⊆

ﬁniteΛ⊆Γ

Λ

scl(intU)

. (3.1)

Since eachU is semiclosed and(_ΓU)∩A= ∅,X−Ais an open set containingB. But noV∈ᐂis contained inX−A. This contradicts (e).

(f)⇒(g). LetA be a closed subset ofX and letΩbe an open ﬁlterbase onXwhose members have nonempty intersection with A. Suppose that(sadΩ)∩A= ∅. Then

{sclF :F ∈Ω}is a family of semiclosed sets such that(_ΩsclF)∩A= ∅. Hence, in view of (f), choose a ﬁniteΛ⊆Ωsuch that(_Λint(sclF))∩A= ∅. Since eachF is open, this means that(_ΛF)∩A= ∅. This contradicts the assumption sinceΩis a ﬁlterbase onXwhose members have nonempty intersection withA.

(g)⇒(h). LetAbe a closed subset ofXand letV be an open neighborhood ofA. Let

Γ be a semiopen cover ofX−A. ThenΓ is a semiopen cover of the closed setX−V. Suppose that there is no finite Λ⊆Γ satisfyingX−V ⊆ΛclU. Thenᐂ= {W :W = (X−V )−Λ⊆ΓclU, Λfinite}is an open filterbase on the closed setX−V. Therefore, in view of (g),(sadᐂ)∩(X−V )= ∅. Hence,(_ᐂsclW )∩(X−V )= ∅. Then(_Γscl(X−

clU))∩(X−V )= ∅. Hence,(X−V )∩(X−Γsint(clU))= ∅. Since eachUis semiopen, U⊆sint(clU). Therefore,(X−ΓU)∩(X−V )= ∅, a contradiction to the fact thatΓ is a cover ofX−V. Therefore,X=(_ΛclU)∪Vfor some ﬁniteΛ⊆Γ.

(g)⇒(a). LetAbe a closed subset of(X,T )and letᐁbe aT-semiopen cover ofA. Assume that there is no ﬁnite ᏿⊆ᐁ satisfyingA⊆᏿clU. Considerᐂ= {V :V =

᏿⊆ᐁ(X−clU), ᏿ﬁnite}. Thenᐂis an open ﬁlterbase onX andV∩A= ∅for each V∈ᐂ. Then proceeding as in the proof of (g)⇒(h), a contradiction to the fact thatᐁis a cover ofAis reached.

(h)⇒(d). LetΩbe an open ﬁlterbase onXand letA=sadΩ. LetUbe an open neigh-borhood ofA. Thenᐂ= {X−sclF:F∈Ω}is a semiopen cover ofX−Aand hence of X−U. Also, consideringU as a neighborhood of X−(X−U), in view of(h), choose a ﬁniteΓ ⊆Ωsuch thatX=U∪(_Γcl(X−sclF)). ThusX=U∪(_Γ(X−int(sclF))). Hence,X⊆U∪(X−ΓF). Choose anF∈Ωsuch thatX=U∪(X−F). ThenF⊆Uand

Ωiss-adherent convergent.

As mentioned earlier, a C-compact space and a sequentially C-compact space [8] are characterized using a family ofH-sets of the space. Below, a characterization of C-s-compact spaces is given in terms of a family of proper subsets which areS-sets.

Theorem_3.3. _{A space}X_{is a}C-s-compact space if and only if every nonempty proper

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Proof. Assume that every nonempty proper closed subset ofXis anS-set. It will

be enough to prove thatX isS-closed. Letᐁbe a semiopen cover ofX. Assume that eachU∈ᐁis proper, semiopen, and nonempty. Choose aU∈ᐁand an open setF inXsuch thatF⊆U⊆clF. Now,X−F is a nonempty proper closed subset ofX and hence is anS-set. Also,ᐁis a semiopen cover ofX−F. Choose a ﬁnite᏿⊆ᐁsuch that X−F⊆᏿clU. ThenX=(᏿clU)∪Uand hence,XisS-closed.

It is obvious that ifXisC-s-compact, every proper closed subset is anS-set.

Corollary _3.4. _{A space} X_is C-s-compact if and only if every closed subset is an

S-set.

Corollary3.5. A spaceXisC-s-compact if and only ifXis compact and extremally

disconnected.

Proof. _{The proof follows easily from [2, Corollary 2.4 and Proposition 4.1] which}

establish that every closed subset of a spaceXis anS-set if and only ifXis compact and extremally disconnected.

Corollary3.6. The following are equivalent for a spaceX:

(b) letAbe a nonempty proper closed subset ofX, then for every ﬁlterbaseΩonA, A∩sadθΩ= ∅;

(c) letAbe a nonempty proper closed subset ofX, then for every open ﬁlterbaseΩ

onA,A∩sadΩ= ∅.

Proof. _{The proof follows from}_{Theorem 3.3}_{and [2, Proposition 2.2].}

In [10], a class of functions was identiﬁed so that a spaceXisC-compact if and only if a function in this class on the spaceXis a closed function. Here, a class of functions is identiﬁed, which characterizesC-s-compact spaces similarly.

Theorem_3.7. _{The following are equivalent for a space}X:

(b) each functiongfromXto a spaceY with a strongly sub-semiclosed inverse is a closed function;

(c) each functiongfromXto a spaceY with a strongly sub-semiclosed inverse is a semiclosed function.

Proof. (a)⇒(b). Let X beC-s-compact, letA be a closed subset of X, and let g:

X→Y have a strongly sub-semiclosed inverse. Lety be a limit point ofg(A). Then

Ω= {g−1_(W_{− {}_y_}₎_∩_A_:_W _∈_Σ_(y)_} _{is a ﬁlterbase on}_{A. In view of}_{Theorem 3.2(b),} A∩sadθΩ= ∅. ButA∩sadθΩ⊆A∩g−1(y)since ghas a strongly sub-semiclosed inverse. Thus,A∩g−1_(y)_{= ∅}_,_y_∈_{g(A), and}_g(A)_{is closed.}

(b)⇒(c). The proof is obvious.

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eachF∈Ω, and consequentlyv∈sclA−AinY. Suppose otherwise and letB∈SO(v) inY. Choose an open setU⊆B⊆clU. Ifv∈U, thenU contains anF ∈Ω. Suppose that v ∈U and B∩F = ∅for some F ∈Ω. For such an F, (F∪ {v})∩U = ∅. So, v ∈clU, a contradiction. Hence, each B∈ SO(v)in Y satisfies B∩F = ∅ for each F ∈Ω. Letg:X→Y be defined asg(x)=x forx∈Aand g(x)=v forx∈X−A. Theng−1_⊆_g(X)_×_X _{has a strongly sub-semiclosed graph for, if}_Λ_{is a filterbase on} g(X)− {y}withΛ→yinY, theny=vand sadθg−1(Λ)⊆sadθΩ⊆g−1(v). So,Ais closed inX,g(A)=A, andv∈sclA−AinY. Thus,gis not semiclosed.

Remark3.8. In view of the facts that a function with a strongly subclosed graph has

a strongly sub-semiclosed graph and every closed function is semiclosed, the following corollary is an improvement of the suﬃciency of the main result of [10] which states that a spaceXisC-compact if and only if each function onXwith a strongly subclosed inverse is a closed function.

Corollary_3.9. _{A space}X_isC-compact if every function onX_{with a strongly}

sub-semiclosed inverse is sub-semiclosed.

Proof. _{The proof is immediate since every}C-s-compact space isC-compact.

Theorem _3.10. _{A space}Y _is C-s-compact if and only if for every spaceX, every

strongly sub-semiclosed relation fromXtoY is u.s.c.

Proof. Suppose thatY isC-s-compact andRis a strongly sub-semiclosed relation

from a spaceXtoY. Letx∈XandW∈Σ(R(x))inY. If{x}is open, then{x} ∈Σ(x) andR({x})⊆W. Suppose that{x}is not open. LetV∈Σ(x)inX. ThenΩ= {V−{x}: V∈Σ(x)}is a ﬁlterbase onX−{x}andΩ→x. Suppose thatR(V−{x})∩(Y−W )= ∅

is satisﬁed for eachV∈Σ(x). ThenΓ= {R(V−{x})∩(Y−W ):V∈Σ(x)}is a ﬁlterbase on the closed subsetY−Wof theC-s-compact spaceY. Hence, in view ofTheorem 3.2, (sadθΓ)∩(Y−W )= ∅.SinceRis a strongly sub-semiclosed relation, sadθR(V−{x})⊆ R(x)and hence, sadθR(V− {x})∩(Y−W )⊆R(x)∩(Y−W ). This is a contradiction sinceR(x)⊆W. Therefore, there is aV∈Σ(x)such thatR(V−{x})⊆WandR(x)⊆W. Hence, there is aV∈Σ(x)such thatR(V )⊆W. Therefore,Ris a u.s.c. relation.

Conversely, suppose thatY is notC-s-compact. Relying onCorollary 3.6(b), choose a nonempty proper closed subsetAofYand a ﬁlterbaseΩonAwithA∩sadθΩ= ∅. Let v∈Y−A. LetXbe the setY with the topology{V⊆X:v∈V , orF⊆Vfor someF∈

Ω}. The spaceXis readily seen to be Hausdorff. Define a relationR⊆X×Y as follows: (x,x)∈Rifx=v;(v,x)∈Rifx∈X−A. The relationR is strongly sub-semiclosed for, ifΓ is a filterbase onX−{x}withΓ→xinX, thenx=v. Also, sadθR(Γ)=sadθΩ⊆ Y−A=R(v). ButRis not u.s.c. sinceY−A∈Σ(R(v))inY, and there is noV∈Σ(v) inXsuch thatR(V )⊆Y−A.

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twoC-s-compact spaces need not beC-s-compact. From the deﬁnitions ofC-s-compact spaces andC-compact spaces and from the fact that every open set is semiopen, it is clear that aC-s-compact space isC-compact. The following example (see [15, Example 2]) is an example of a HausdorfC-compact space which is not compact and hence not C-s-compact.

Example4.1. LetNrepresent the set of positive integers. LetY denote the subset

of the plane consisting of all points of the form(1/n,1/m)and points of the form (1/n,0)fornand minN. LetX=Y∪ {∞}. TopologizeX as follows: let each point (1/n,1/m)be open. PartitionNinto inﬁnitely many inﬁnite equivalence classes{Ni}∞i=1. A neighborhoodU(1/i,0)of(1/i,0)is of the form{(1/i,0)} ∪ {(1/i,1/m):m≥k} ∪ {(1/n,1/m):m∈Ni, n≥k}for somek∈N. A neighborhood system for the point∞ consists of all sets of the formX−T, where

T=

₁

n,0

:n∈N ∪  k

i=1

₁

i, 1 m

:m∈N ∪

₁

n, 1 m

:m∈Ni, n∈N 



(4.1)

for somek∈N. The spaceXwith this topology is Hausdorff and is not compact. Hence, it is not C-s-compact. It can also be verified that for the closed setG= {(i/n,0): n∈N}and the semiopen cover ᐁ= {Un:Un= {(1/n,1/m):m≥k for somek∈ N}∪{(1/n,0)}}ofG, there is no finite᏿⊆ᐁsuch thatG⊆᏿clU.

The following example illustrates that, in general, a compact space need not be C-s-compact.

Example_4.2. _Let X=∞_i

=0Xi, whereX0= {(0,1/n):n∈N} ∪ {(0,0)} and Xi =

{(1/i,1/n):n∈N}∪{(1/i,0)}fori≥1. Fori≥1, each point ofXi, except(1/i,0), is open. Basic open subsets

U1 i,0

=1_i,0 ∪1_i,1 n

:n > jfor somej∈N . (4.2)

Basic open subsetsU(0,1/i)= {(0,1/i)}∪{(1/n,1/i)for all but finitely manyn∈N}. A basic open set containing(0,0)contains all the points on all but finitely manyXi, i∈N. ThenX is compact since an open set containing(0,0)contains all the points on all but finitely manyXi’s, sayXp1,Xp2,...,Xpn. For eachi∈ {1,2,...,n},U(1/Pi,0) contains all but finitely many points onXpi. The spaceXis notC-s-compact. To prove this, consider the closed setA= {(1/n,0):n∈N} ∪ {(0,0)}. Consider the semiopen cover

ᐁ=

₁

2n,0 ∪

₁

2n, 1 m

:n,m∈N ∪{(0,0)}

∪

₁

(2n−1),0 ∪

₁

(2n−1), 1 m

:m > j

for somej∈N:n,m∈N .

(4.3)

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An example of anS-closed space which is notC-s-compact is given as follows.

Example4.3. LetDbe the set of positive integers with discrete topology. Then the

Katˇetov expansionkDis extremally disconnected andH-closed and hence isS-closed. But it is not compact and hence notC-s-compact.

In a class of Hausdorﬀ spaces, the following implication diagram is valid, no implication is reversible, in general, other than the following one:

compact + ED compact

C-s-compact C-compact

S-closed.

(4.4)

The following lemma is a corollary to [13, Theorem 7].

Lemma4.4. For a compact Hausdorﬀ space, the following are equivalent:

(i) Xis extremally disconnected; (ii) Xis projective;

(iii) XisS-closed.

The product of twoC-s-compact spaces need not beC-s-compact as can be seen from the following example.

Example_4.5. _{The Stone-Cech compactiﬁcation}βN_{of the set of natural numbers}

is compact, Hausdorﬀ, and extremally disconnected and hence is C-s-compact. But βN×βNis compact, Hausdorﬀ, and not extremally disconnected, and hence it is not S-closed in view of the above lemma. Therefore, it is notC-s-compact.

Remark _4.6. _{In fact, if} D _{is an inﬁnite discrete space, the Stone-Cech}

compacti-ficationβDis compact, Hausdorff, and extremally disconnected. But βD×βDis not extremally disconnected, but is compact and Hausdorff.

Acknowledgment. This work was partially supported by Housing and Urban

De-velopment HUD Special Projects Grant B 98 SP MD 0074.

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Bhamini M. P. Nayar: Department of Mathematics, Morgan State University, Baltimore, MD 21251, USA