© Hindawi Publishing Corp.
ON
S
-CLUSTER SETS AND
S
-CLOSED SPACES
M. N. MUKHERJEE and ATASI DEBRAY
(Received 15 July 1996 and in revised form 3 November 1997)
Abstract.A new type of cluster sets, calledS-cluster sets, of functions and multifunc-tions between topological spaces is introduced, thereby providing a new technique for studyingS-closed spaces. The deliberation includes an explicit expression ofS-cluster set of a function. As an application, characterizations of Hausdorff andS-closed topological spaces are achieved via such cluster sets.
Keywords and phrases.S-cluster set,S-closedness,θs-closure, semi-open sets.
2000 Mathematics Subject Classification. Primary 54D20, 54D30, 54C60, 54C99.
1. Introduction. The theory of cluster sets was developed long ago, and was ini-tially aimed at the investigations of real and complex function theory (see [15]). A com-prehensive collection of works in this direction can be found in the classical book of Collingwood and Lohwater [1]. Weston [14] was the first to initiate the corresponding theory for functions between topological spaces basically for studying compactness. Many others (e.g., see [3, 4, 6]) followed suit with cluster sets,θ-cluster sets andδ -cluster sets of functions and multifunctions, ultimately implicating different covering properties, among other things.
The present paper is intended for the introduction of a new type of cluster sets, calledS-cluster sets, which provides a new technique for the study ofS-closedness of topological spaces. It is shown that such cluster sets of suitable function can charac-terize Hausdorffness. Finally, we achieve, as our prime motivation, certain character-izations of anS-closed space.
In what follows,X and Y denote topological spaces, andf :X→Y is a function fromXintoY. By a multifunctionF :X→Y we mean, as usual, a function mapping points ofX into the nonempty subsets ofY. The set of all open sets of (X,τ), each containing a given pointxofX, is denoted byτ(x). A setA(⊆X)is called semi-open [5] if for some open setU,U⊆A⊆clU, where clUdenotes the closure ofUinX. The set of all semi-open sets ofX, each containing a given subsetAofX, is denoted by SO(A), in particular, ifA= {x}, we write SO(x)instead of SO({x}). The complements of semi-open sets are called semi-closed. For any subsetAofX, theθ-closure [13] (θ-semiclosure [8]) ofA, denoted byθ-clA(respectively,θs-clA), is the set of all points
xofXsuch that for everyU∈τ(x)(respectively,U∈SO(x)), clU∩A≠∅. The set
Ais called θ-closed [13] (θ-semiclosed [8]) if A=θ-clA(respectively, A=θs-clA). It is known [9] thatθ-clAneed not beθ-closed, but it is so ifAis open. A nonvoid collectionΩof nonempty subsets of a spaceXis called a grill [12] if
(ii) A∪B∈Ω⇒A∈ΩorB∈Ω.
A filterbaseᏲ on a spaceXis said toθS-adhere [11] at a pointx ofX, denoted as
x∈θS-adᏲ, ifx∈ ∩{θs-clF:F∈Ᏺ}. A grillΩonXis said toθs-converge to a point
xofX, if to eachU∈SO(x), there corresponds someG∈ΩwithG⊆clU. A setAin a spaceXis said to beS-closed relative toX[7] if for every coverᐁofAby semi-open sets ofX, there exists a finite subfamilyᐁ0ofᐁsuch thatA⊆ ∪{clU:U∈ᐁ0}. If, in addition,A=X, thenXis called anS-closed space [11].
2. Main theorem and associated results. We begin by introducingS-cluster set of a function and of a multifunction between two topological spaces.
Definition2.1. LetF :X→Y be a multifunction andx∈X. Then theS-cluster set ofFatx, denoted byS(F,x), is defined to be the set∩{θ-clF(clU):U∈SO(x)}. Similarly, for any functionf:X→Y, theS-cluster setS(f ,x)off atx is given by ∩{θ-clf (clU):U∈SO(x)}.
In the next theorem, we characterize theS-cluster sets of functions between topo-logical spaces.
Theorem2.2. For any functionf:X→Y, the following statements are equivalent. (a)y∈S(f ,x).
(b)The filterbasef−1(clτ(y))θS-adheres atx.
(c)There is a grillΩonXsuch thatΩθs-converges toxandy∈ ∩{θ-clf (G):G∈Ω}. Proof. (a)⇒(b). y∈S(f ,x)⇒ for eachW∈SO(x)and eachV ∈τ(y), clV∩
f (clW )≠∅ ⇒for eachW ∈SO(x)and eachV∈τ(y), f−1(clV )∩clW≠∅. This ensures that the collection{f−1(clV ):V ∈τ(y)}(which can easily be seen to be a filterbase onX)θS-adheres atx.
(b)⇒(c). LetᏲbe the filter generated by the filterbasef−1(clτ(y)). ThenΩ= {G⊆
X:G∩F≠∅,for eachF∈Ᏺ}is a grill onX. By the hypothesis, for eachU∈SO(x)
and eachV ∈τ(y), clU∩f−1(clV)≠∅. Hence,F∩clU≠∅for eachF ∈Ᏺ and eachU∈SO(x). Consequently, clU∈Ωfor allU∈SO(x), which proves thatΩθs -converges tox. Now, the definition ofΩyields thatf (G)∩clW≠∅for allW∈τ(y)
and allG∈Ω, i.e.,y∈θ-clf (G)for allG∈Ω. Hence,y∈ ∩{θ-clf (G):G∈Ω}. (c)⇒(a). LetΩbe a grill onXsuch thatΩθs-converges tox, andy∈ ∩{θ-clf (G):
G∈Ω}. Then {clU:U∈SO(x)} ⊆Ωand y∈θ-clf (G)for eachG∈Ω. Hence, in particular,y∈θ-clf (clU)for allU∈SO(x). So,y∈ ∩{θ-clf (clU):U∈SO(x)} =
S(f ,x).
In what follows, we show thatS-cluster sets of a function may be used to ascertain the Hausdorffness of the codomain space.
Theorem2.3. Letf:X→Ybe a function on a topological spaceXonto a topological spaceY. ThenY is Hausdorff ifS(f ,x)is degenerate for eachx∈X.
Proof. Lety1,y2∈Ysuch thaty1≠y2. Asfis a surjection, there arex1,x2∈X such thatf (xi)=yifori=1,2. Now, sinceS(f ,x)is degenerate for eachx∈X,y2=
∅, i.e.,f (clU)⊆Y−clV. Then the open setsY−clVandVstrongly separatey1and
y2inY, which proves thatY is Hausdorff.
Remark2.4. We note that the converse of the above theorem is false. For example, consider the identity mapf:(R,τ)→(R,σ ), whereτandσ, respectively, denote the cofinite topology and the usual topology on the setRof real numbers. ThenS(f ,x)=
R, for eachx∈R, though(R,σ )is aT5-space.
In order to obtain the converse, we introduce the following class of functions. Definition2.5. A functionf:X→Y is calledθs-irresolute onXif for eachx∈X
and each semi-open setV containingf (x), there is a semi-open setU containingx
such thatf (clU)⊆V.
Theorem2.6. Letf:X→Y be aθs-irresolute function withY a Hausdorff space. ThenS(f ,x)is degenerate for eachx∈X.
Proof. Letx∈X. Asf isθs-irresolute onX, for anyV∈SO(f (x)), there isU∈ SO(x)such thatf (clU)⊆V. ThenS(f ,x)= ∩{θ-clf (clU):U∈SO(x)} ⊆ ∩{θ-clV:
V∈SO(f (x))}. Lety∈Y withy≠f (x). AsY is Hausdorff, there are disjoint open setsU,W withy∈U,f (x)∈W. Obviously, asU∩clW= ∅,y∉clW=θ-clW. As
W∈τ(f (x))⊆SO(f (x)),y∉∩{θ-clV:V∈SO(f (x))}and hencey∉S(f ,x). Thus,
S(f ,x)= {f (x)}.
Combining the last two results, we get the following characterization for the Hausdorffness of the codomain space of a kind of function in terms of the degen-eracy of itsS-cluster set.
Corollary2.7. Letf:X→Y be aθs-irresolute function on Xonto Y. Then the spaceY is Hausdorff if and only ifS(f ,x)is degenerate for eachxofX.
We have just seen that degeneracy of theS-cluster set of an arbitrary function is a sufficient condition for the Hausdorffness of the codomain space. We thus like to ex-amine some other situations when theS-cluster sets are degenerate, thereby ensuring the Hausdorffness of the codomain space of the function concerned. To this end, we recall that a topological space(X,τ)is almost regular [10] if for every regular closed setAinXand for eachx∉A, there exist disjoint open setsUandVsuch thatx∈U
andA⊆V. It is known that in an almost regular spaceX,θ-clAisθ-closed for each
A⊆X. A functionf :X→Y carryingθ-closed sets ofX intoθ-closed sets ofY is called aθ-closed function [2].
Theorem2.8. Letf:X→Y be aθ-closed map from an almost regular space into a spaceY. I ff−1(y)isθ-closed inXfor ally∈Y, thenS(f ,x)is degenerate for each
x∈X.
Proof. We haveS(f ,x)= ∩{θ-clf (clU):U ∈SO(x)} ⊆ ∩{θ-clf (θ-clU):U∈ SO(x)}. AsX is almost regular, θ-clU is θ-closed for all U∈SO(x). Now, sincef
y∉f (clP)=f (θ-clP)(asPis an open set) and, hence,y∉∩{f (θ-clU):U∈SO(x)}. In view of what we have deduced above, we conclude thaty∉S(f ,x), which proves thatS(f ,x)is degenerate.
Theorem2.9. Letf:X→Xbe aθ-closed injection on an almost regular Hausdorff spaceXintoY. ThenS(f ,x)is degenerate for eachx∈X.
Proof. AsXis almost regular andf is aθ-closed map, we haveθ-clf (θ-clU)=
f (θ-clU)for anyU∈SO(x)and, hence,
S(f ,x)= ∩{θ-clf (clU):U∈SO(x)} ⊆ ∩{θ-clf (θ-clU):U∈SO(x)}
= ∩{f (θ-clU):U∈SO(x)}. (2.1)
Forx,x1∈Xwithx≠x1,f (x)≠f (x1)asf is injective. By the Hausdorffness ofX, there are disjoint open setsU,V inX withx∈U,x1∈V. Obviously,U∩clV= ∅. SO,x1∉θ-clUand hencef (x1)∉f (θ-clU). SinceU∈τ(x)⊆SO(x), equation (2.1) yieldsf (x1)∉S(f ,x). Thus,S(f ,x)is degenerate for eachx∈X.
The above theorem is equivalent to the following apparently weaker result whenX
is regular.
Theorem2.10. Iff:X→Y is aθ-closed injection on aT3spaceXinto a spaceY, thenS(f ,x)is degenerate for eachx∈X.
Proof. It is known that in a regular spaceX,θ-clU=clU for anyU⊆X. Since
X isT3and f is a θ-closed injection,{f (x)} ⊆S(f ,x)= ∩{f (clU):U∈SO(x)} ⊆ ∩{f (clU):U∈τ(x)} = {f (x)}.
Note that the above result is indeed equivalent to that of Theorem 2.9 follows from the following considerations: for any subsetAof a topological space(X,τ),θ-closure ofAin(X,τ)is the same as that in(X,τs), where(X,τs)denotes the semiregulariza-tion space [9] of(X,τ). Moreover, it is known [9] that(X,τ)is Hausdorff (almost reg-ular) if and only if(X,τs)is Hausdorff (respectively, regular). Now, since SO(X,τs)⊆ SO(X,τ), it follows that S(f ,x)=S(f :(X,τ)→Y ,x)⊆S(f :(X,τs)→Y ,x). So,
S(f ,x)is degenerate for eachx∈X if(X,τ)is an almost regular Hausdorff space andf:X→Y is aθ-closed injection.
A sort of degeneracy condition for theS-cluster set of a multifunction withθ-closed graph is now obtained.
Theorem2.11. For a multifunction F : X →Y, if F has a θ-closed graph, then
S(F,x)=F(x).
Proof. For anyy∈S(F,x), clW∩F(clU)≠∅and henceF−(clW )∩clU≠∅for
eachU∈SO(x)and eachW ∈τ(y), where, as usual,F−(B)= {x∈X:F(x)∩B≠
∅} for any subsetB ofY. Then for any basic open setM×N in X×Y containing
(x,y),F−(clN)∩clM≠∅. So,(clM×clN)∩G(F)≠∅and hence cl(M×N)∩G(F)≠
∅, whereG(F)= {(x,y)∈X×Y :y∈F(x)}denotes the graph ofF. So,(x,y)∈
θ-clG(F)=G(F) (asG(F) is θ-closed). Hence, (x,y)∈[G(F)∩({x} ×Y )] so that
It is obvious thatF(x)⊆S(F,x)for eachx∈X. Hence,S(F,x)=F(x)holds for all
x∈X.
The next result serves as a partial converse of the above one.
Theorem2.12. For a multifunctionF :X→Y, ifS(F,x)=F(x) for eachx∈X, then the graphG(F)ofFisθ-semiclosed (and hence semi-closed).
Proof. Let(x,y)∈X×Y−G(F). Now,y ∉F(x)=S(F,x)⇒there exist some
W∈SO(x)and someV∈τ(y)such that clV∩F(clW )= ∅ ⇒(clW×clV)∩G(F)= ∅ ⇒cl(W×V)∩G(F)= ∅. AsW×V is a semi-open set inX×Y containing(x,y),
(x,y)∉θs-clG(F). Hence,G(F)isθ-semi-closed.
We now turn our attention to the characterizations ofS-closedness viaS-cluster sets. We need the following lemmas for this purpose.
Lemma2.13. A setAin a topological spaceXis anS-closed set relative toXif and only if for every filterbaseᏲonXwithF∩C≠∅for allF∈Ᏺand for allC∈SO(A),
A∩θS-adᏲ≠∅.
Proof. LetAbe anS-closed set relative toXand letᏲbe a filterbase onXwith the stated property. If possible, suppose thatA∩θS-adᏲ= ∅. Then for eachx∈A, there is a semi-open setV(x)inX containingxsuch that cl(V (x))∩F(x)= ∅for someF(x)∈Ᏺ. Now,{V(x):x∈A}is a cover ofAby semi-open sets ofX. By theS -closedness ofArelative toX, there is a finite subsetA∗ofAsuch thatA⊆ ∪{clV (x): x∈A∗}. ChooseF∗∈Ᏺ such thatF∗⊆ ∩{F(x):x∈A∗}. Then F∗∩(∪{clV(x): x∈A∗})= ∅, i.e.,F∗∩cl(∪{V(x):x∈A∗})= ∅. Now, as ∪{V (x):x∈A∗}is a
semi-open set inX,∪{clV(x):x∈A∗} ∈SO(A), a contradiction.
Conversely, assume thatAis notS-closed relative toX. Then for some cover{Uα:
α∈Λ}ofAby semi-open sets ofX,Aα∈Λ0clUαfor each finite subsetΛ0ofΛ.
So,Ᏺ= {A−α∈Λ0clUα:Λ0is a finite subset ofΛ}is filterbase onX, withF∩C≠∅,
for eachF∈Ᏺand eachC∈SO(A). ButA∩θS-adᏲ= ∅.
Lemma2.14[8, 11]. (a)A topological spaceXisS-closed if and only if every filter-baseθS-adheres inX.
(b)Anyθ-semiclosed subset of anS-closed spaceXisS-closed relative toX.
Definition2.15. For a function or a multifunctionF:X→Y and a setA⊆X, the notationS(F,A)stands for the set∪{S(F,x):x∈A}.
Theorem2.16. For any topological spaceX, the following statements are equiva-lent.
(a)XisS-closed.
(b)S(F,A)⊇ ∩{θ-clF(U):U∈SO(A)}for eachθ-semiclosed subsetAofX, for each topological spaceY and each multifunctionF:X→Y.
(c)S(F,A)⊇ ∩{θs-clF(U):U∈SO(A)}for eachθ-semiclosed subsetAofX, for each topological spaceY and each multifunctionF:X→Y.
Then for allW∈τ(z)and for eachU∈SO(A), clW∩F(U)≠∅, i.e.,F−(clW )∩U≠∅.
Thus,Ᏺ= {F−(clW ):W∈τ(z)}is clearly a filterbase onX, satisfying the condition
of Lemma 2.13. Hence,x∈A∩θS-adᏲ. Thenx∈A, and for allU∈SO(x)and each
W∈τ(z), clU∩F−(clW )≠∅, i.e.,F(clU)∩clW≠∅ ⇒z∈S(F,x)⊆S(F,A).
(b)⇒(c). Obvious.
(c)⇒(a). In order to show that X is S-closed, it is enough to show, by virtue of Lemma 2.14(a), that every filterbaseᏲ onX θS-adheres at somex∈X. LetᏲbe a filterbase onX. Takey0∉X, and constructY =X∪ {y0}. Define,τY= {U⊆Y :y0∉
U} ∪ {U⊆Y :y0∈U,F⊆Ufor someF∈Ᏺ}. ThenτY is a topology onY. Consider the function α:X →Y by α(x)=x. In order to avoid possible confusion, let us denote the closure andθs-closure of a setAinX(Y ), respectively, by clXA(clYA)and θs-clXA(respectively,θs-clYA). AsX is θ-semiclosed inX, by the given condition, S(α,X)⊇ ∩{θs-clYα(U): U ∈SO(X)} = ∩{θs-clYU :U ∈SO(X)} =θs-clYX. We
considery0∈Y andP0∈SO(y0). There is someW∈τY such thatW⊆P0⊆clYW.
If y0 ∉W, then W ⊆ X and hence clYW∩X ≠∅. If on the other hand, y0∈ W, then there is someF ∈Ᏺ such that F ⊆W, i.e., clYF ⊆clYW. So, X∩clYW ≠∅.
So, in any case,X∩clYW≠∅and, consequently, as clYW =clYP0, X∩clYP0≠∅. Thus,y0∈θs-clYX. So,y0∈S(α,x)for somex∈X. Consider anyV∈SO(x)and
F∈Ᏺ. ThenF∪ {y0} ∈τY. Again,Y−(F∪ {y0})is a subset ofY not containingy0. Thus,Y−(F∪ {y0})is open inY, which proves that clY(F∪ {y0})=F∪ {y0}. Now,
clXV∩F=α(clXV )∩clY(F∪{y0})≠∅. Thus,x∈θS-adᏲ.
Acknowledgement. The authors are grateful to the referee for his sincere eval-uation and some constructive comments which improved the paper considerably.
References
[1] E. F. Collingwood and A. J. Lohwater,The Theory of Cluster Sets, Cambridge University Press, Cambridge, 1966. MR 38#325. Zbl 149.03003.
[2] A. J. D’Aristotile, On θ-perfect mappings, Boll. Un. Mat. Ital. (4) 9 (1974), 655–661. MR 51#1710.
[3] T. R. Hamlett,Cluster sets in general topology, J. London Math. Soc. (2)12(1975/76), no. 2, 192–198. MR 52#9148. Zbl 317.54010.
[4] J. E. Joseph,Multifunctions and cluster sets, Proc. Amer. Math. Soc.74(1979), no. 2, 329– 337. MR 80f:54011. Zbl 405.54014.
[5] N. Levine,Semi-open sets and semi-continuity in topological spaces, Amer. Math. Monthly 70(1963), 54–61. MR 29#4025. Zbl 113.16304.
[6] M. N. Mukherjee and S. Raychaudhuri,A new type of cluster sets and its applications, Tamkang J. Math.26(1995), no. 4, 327–336. MR 97g:54033. Zbl 869.54022. [7] T. Noiri,OnS-closed spaces, Ann. Soc. Sci. Bruxelles Sér. I 91(1977), no. 4, 189–194.
MR 57#13851. Zbl 408.54017.
[8] ,OnS-closed spaces andS-perfect functions, Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur.120(1986), no. 3-4, 71–79 (1987). MR 89g:54058. Zbl 794.54028.
[9] J. R. Porter and R. G. Woods,Extensions and Absolutes of Hausdorff Spaces, Springer-Verlag, New York, 1988. MR 89b:54003. Zbl 652.54016.
[10] M. K. Singal and S. Prabha Arya,On almost-regular spaces, Glasnik Mat. Ser. III4 (24) (1969), 89–99. MR 39#4804. Zbl 169.24902.
[12] W. J. Thron,Proximity structures and grills, Math. Ann.206(1973), 35–62. MR 49#1483. Zbl 256.54015.
[13] N. V. Veliˇcko,H-closed topological spaces, Amer. Math. Soc. Transl.78(1968), 103–118. [14] J. D. Weston,Some theorems on cluster sets, J. London Math. Soc.33(1958), 435–441.
MR 20#7109. Zbl 089.17501.
[15] W. H. Young,La Syme’tric de structure des fonctions de variables reelles, Bull. Sci. Math. 52(1928), no. 2, 265–280.