S. S. BENCHALLI, B. M. ITTANAGI, AND P. G. PATIL
Received 27 September 2005; Revised 10 May 2006; Accepted 30 May 2006
J. M. Aarts introduced and studied a new dimension function, Hind, in 1975 and obtained several results on this function. In this paper, a new local inductive dimension function called local huge inductive dimension function denoted by loc Hind is introduced and studied. Furthermore, an effort is made to introduce and study dimension functions for fuzzy topological spaces. It has been possible to introduce and study the small inductive dimension function indfX and large inductive dimension function IndfX for a fuzzy topological spaceX.
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1. Introduction
InSection 2, a new local dimension function called local huge inductive dimension func- tion denoted by loc Hind is introduced and studied. Its relationship with other local di- mension functions is established. A closed subset theorem and an open subset theorem are obtained for the local huge inductive dimension function. Further it is also proved that the local huge inductive dimension function coincides with the huge inductive di- mension function for the class of weakly paracompact totally normal spaces.
The concept of a fuzzy subset was introduced and studied by Zadeh [9] and the con- cept of fuzzy topological spaces by Chang [2]. Many mathematicians have contributed to the development of fuzzy topological spaces.
InSection 3, two inductive types of dimension functions for fuzzy topological spaces have been introduced and studied. Several results have been obtained. It is observed that such dimensions are integers and not fractions.
2. A new local dimension function for topological spaces The following definition is due to Aarts [1].
Definition 2.1 [1]. The huge inductive dimension function Hind is defined for every hereditarily normal space as follows. HindX= −1 if and only if X=φ. For each in- teger n≥0, HindX≤n provided that for each pair of closed subsets F and G with
Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences Volume 2006, Article ID 92870, Pages1–8
DOI10.1155/IJMMS/2006/92870
Hind(F∩G)≤n−1 there exists a pair of closed subsetsKandLsuch thatF−G⊂K−L, G−F⊂L−K,K∪L=X, and Hind(K∩L)≤n−1. HindX=nif and only if Hind X≤nis true and HindX≤n−1 is not true. HindX= ∞if and only if HindX≤nis not true for everyn.
The following result is due to Aarts [1].
Proposition 2.2 [1]. For each integern≥0, HindX≤nif and only if for each pair of closed subsetsFandGwith Hind(F∩G)≤n−1 there exists a closed setSsuch thatF−G andG−Fare separated bySand HindS≤n−1. That is,X−S=U∪V, whereU,V are disjoint open sets inX,F−G⊂U,G−F⊂V, and HindS≤n−1.
The following concept of a barrier is due to Va˘ınˇste˘ın [6].
Definition 2.3 [6]. LetA,Bbe a pair of closed subsets of a spaceX. Then a closed subset CofXis said to be a barrier betweenAandBifX−[C∪(A∩B)]=G∪H, whereG,H are disjoint open sets inXsuch thatA−B⊂GandB−A⊂H.
The following result is proved.
Theorem 2.4. LetXbe a hereditarily normal space. If, for any two closed setsA,BinXwith Hind (A∩B)≤n−1, there is a barrierCbetweenAandBinXsuch that HindC≤n−1, then HindX≤n.
Proof. By hypothesis, there is a barrier CbetweenA andB such that HindC≤n−1.
SinceCis a barrier betweenAandB,X−[C∪(A∩B)]=G∪H, whereG,Hare disjoint open sets inX such thatA−B⊂G andB−A⊂H. Clearly the closed setC∪(A∩B) separatesA−BandB−A. Also HindC≤n−1 and Hind(A∩B)≤n−1. Therefore by the countable sum theorem for the huge inductive dimension [1, Theorem 1] it follows that Hind[C∪(A∩B)]≤n−1. Hence byProposition 2.2[1, Proposition 1] it follows
that HindX≤n.
The concept of local dimension for the huge inductive dimension function is intro- duced in the following.
Definition 2.5. The local huge inductive dimension, loc Hind, is defined for every hered- itarily normal spaceX as follows. loc HindX= −1 if and only ifX=φ.loc HindX≤n if and only if for each pointx∈X there exists an open setU containingx such that HindU≤n. loc HindX=n, forn=0, 1, 2,...,∞, are defined as usual.
Remark 2.6. If loc HindX≤n,x∈X, andUis an open set containingx, then there exists an open setVinXsuch thatx∈V⊂Uand HindV≤n. Thus, for a hereditarily normal spaceX, loc HindX≤n if and only if every open cover ofX has an open refinement {Uλ:λ∈Λ}such that HindUλ≤nfor eachλ∈Λ.
The relationships of loc Hind with some of the other dimension functions are obtained in the following.
Theorem 2.7. For a hereditarily normal spaceX, the following statements are true.
(i) loc HindX≤HindX.
(ii) loc dimX≤loc IndX≤loc HindX.
Proof. (i) Let Hind X≤n. Then every point x∈X has the neighborhoodX such that HindX=HindX≤n. Therefore loc HindX≤n.
(ii) Since X is hereditarily normal, it is normal, and hence loc dim X≤loc Ind X.
It is required to show that loc IndX≤loc HindX. Let loc HindX≤n. Then each point x∈Xhas a neighborhoodU inX with HindU≤n. NowU⊂X andXis hereditarily normal. ThereforeUis also hereditarily normal. Then from [1, Proposition 2] it follows that IndU≤HindU. Therefore IndU≤nand hence loc HindX≤n. Hence loc IndX≤
loc HindX.
Corollary 2.8. IfXis hereditarily normal regular space, then indX≤loc HindX.
Proof. The result follows since indX≤loc IndXon the class of regular spaces [5].
The closed subset theorem for local huge inductive dimension function is obtained, which is contained in the following.
Theorem 2.9. IfAis a closed subset of a hereditarily normal spaceX, then loc HindA≤ loc HindX.
Proof. Let loc HindX≤n. Let x∈A. Now x∈X and loc HindX≤n. Therefore there exists an open setU inXsuch thatx∈U and HindU≤n. ThenU∩Ais an open set inAcontainingx. NowclA(U∩A) is a closed subset ofAand HindU≤n. Then by the closed subset theorem for Hind [1, Proposition 3] it follows that Hind[clA(U∩A)]≤n.
Therefore loc HindA≤n. Hence loc HindA≤loc HindX.
The open subset theorem for loc Hind is proved in the following.
Theorem 2.10. IfX is a hereditarily normal regular space andY is an open subset ofX, then loc HindY≤loc HindX.
Proof. Let loc HindX≤n. Let y∈Y. Then y∈X and loc HindX≤n. Therefore there exists an open setU in X containing y such that HindU≤n. NowU∩Y is an open set inX containing yandX is regular. Therefore there exists an open set V such that y∈V ⊂V ⊂U∩Y. ThenV is an open neighborhood ofyin Y andV is the closure of V in Y. Since V⊂U, it follows that HindV ≤n. Therefore loc HindX≤n. Hence
loc HindY≤loc HindX.
The next result is a sum theorem that is obtained for loc Hind.
Theorem 2.11. If a hereditarily normal spaceXis the union of two closed setsA,Band if loc HindA≤nand loc HindB≤n, then loc HindX≤n.
Proof. Letx∈X. Ifx∈X−A, thenx∈Band loc HindB≤n. Therefore there exists an open setU∩BinBcontainingx, whereUis open inX such that HindU∩B≤n. Let W=U∩(X−A). ThenWis an open set inXcontainingxsuch thatW⊂U∩Band HindU∩B≤n. Therefore HindW≤nand hence loc HindX≤n. Similarly ifx∈X−B,
then there exists an open set inXcontainingx, the closure of which has huge inductive dimension not exceedingn. Ifx∈A∩B, thenx∈A andx∈B. Since loc HindA≤n and loc HindB≤n, there exist open setsU∩AandV∩BinAandB, respectively, each containingx such that HindU∩A≤nand HindV∩B≤n. LetW=[X−(A−U)]− (B−V). ThenW is an open set containingx andW⊂(U∩A)∪(V∩B). Therefore W⊂(U∩A)∪(V∩B).
Therefore
HindW≤Hind(U∩A)∪(V∩B)
≤supHind(U∩A), Hind (V∩B) (2.1) by using the countable sum theorem for Hind [1]. Therefore HindX≤n. Hence loc Hind
X≤n.
The above result can be extended to a finite family and furthermore to a locally finite family which is contained in the following corollary.
Corollary 2.12. If{Aλ:λ∈Λ}is a locally finite closed covering of a hereditarily normal spaceXsuch that loc HindAλ≤nfor eachλ∈Λ, then loc HindX≤n.
Proof. The straightforward proof is omitted.
The following result shows that loc Hind coincides with Hind on the class of weakly paracompact totally normal spaces.
Theorem 2.13. If X is a weakly paracompact totally normal space, then loc HindX= HindX.
Proof. SinceXis totally normal, it is hereditarily normal, and hence fromTheorem 2.7it follows that loc HindX≤HindX.
On the other hand, sinceX is weakly paracompact totally normal, loc IndX=IndX from [5, Page 197]. But Ind=Hind on the class of totally normal spaces. Therefore HindX=loc IndX≤loc HindX, since for hereditarily normal spaces loc Ind≤loc Hind fromTheorem 2.7. Therefore HindX≤loc HindX. Hence loc HindX=HindX. 3. Inductive dimension functions for fuzzy topological spaces
The main purpose of this section is to introduce and study dimension functions on fuzzy topological spaces. It has been possible to introduce the small inductive dimension func- tion, indfX, and the large inductive dimension function, Indf X, for a fuzzy topological spaceX. A subset theorem is obtained for indfX. It is proved that ifXis a fuzzy topologi- cal space such that IndfX=0 thenXis a normal fuzzy topological space. A closed subset theorem for Indf is also obtained.
The following concept is due to Zadeh [9].
Definition 3.1 [9]. A fuzzy subsetAin a setXis a functionA:X→[0, 1].
The elementary properties related to fuzzy sets are contained in [9]. The fuzzy topo- logical spaces were introduced and studied by Chang [2].
Definition 3.2 [2]. LetXbe a set and letTbe family of fuzzy subsets ofX. ThenTis called a fuzzy topology onXifTsatisfies the following conditions.
(i) 0, 1∈T.
(ii) If{Gλ:λ∈Λ} ⊂T, then∨Gλ∈T.
(iii) IfG,H∈T, thenG∧H∈T.
The pair (X,T) is called a fuzzy topological space (abbreviated as fts). The members ofTare called open fuzzy sets. A fuzzy setBis called a closed fuzzy set if 1−Ais an open fuzzy set.
The concept of the boundary of a fuzzy subset was introduced and studied by Warren [7], which is contained in the following.
Definition 3.3 [7]. LetAbe a fuzzy set in an ftsX. The fuzzy boundary ofAdenoted by bd(A) is defined as the infimum of all the closed fuzzy setsDinX with the following property.D(x)≥A(x) allx∈Xfor which (A∧1−A)(x)>0.
The following results of Warren [7] are used in the sequel.
Theorem 3.4 [7]. LetAandBbe fuzzy sets in an ftsX. Then the following results hold good.
(1) bd(A)=0 if and only ifAis open, closed, and crisp.
(2) bd(A∧B)≤bd(A)∨bd(B).
The other elementary concepts, results, and developments on fuzzy topological spaces can be found in [2–4,7,8].
A new inductive dimension function for fuzzy topological spaces is introduced in the following.
Definition 3.5. LetXbe a fuzzy topological space. The small inductive dimension ofX, denoted by indfX, is defined as follows. indfX= −1 ifX=φ. For any nonnegative inte- gern, indfX≤nif for eachx∈Xand each open fuzzy setGsuch thatG(x)>0 there exists an open fuzzy setUinXsuch thatU(x)>0,U≤Gand indf bd(U)≤n−1. indfX=nif indfX≤nis true and indfX≤n−1 is not true. indfX= ∞if there is no integer n such that indfX≤n.
Note that if X is a general topological space, then this concept reduces to that of ind.
A subset theorem for indf is proved in the following.
Theorem 3.6. IfAis a crisp subset of an ftsX, then indfA≤indfX.
Proof. This is proved by induction onn. Forn= −1, if indfX≤ −1, then indfX= −1, so thatX=φ. SinceAis a crisp subset ofX, it follows thatA=φ, and therefore indfA= −1, that is, indfA≤ −1. Thus if indfX≤ −1, then indfA≤ −1. Therefore the result holds for n= −1. Assume the result for n−1. Then, to prove the result forn, that is, to prove if indfX≤n, then indfA≤n, let indfX≤n. Then to prove indfA≤n, letx∈Aand letG be an open fuzzy set inA, such thatG(x)>0. SinceG is open inA by induced fuzzy topology onA[8], there exists an open fuzzy setHinXsuch thatG=A∧H. Now G(x)>0 impliesH(x)>0 andA(x)>0. Since indfX≤n,His an open fuzzy set inXsuch thatH(x)>0. ByDefinition 3.5there exists an open fuzzy setV inXsuch thatV(x)>0, V ≤H, and indf bd(V)≤n−1. Let U=A∧V. SinceV is an open fuzzy set inX, it
follows thatUis an open fuzzy set inA. NowU(x)>0. We haveA(x)>0 andV(x)>0.
ThereforeA(x)∧V(x)>0, so that (A∧V)(x)>0, and henceU(x)>0. AlsoU≤G. We haveV≤H. ThereforeA∧V≤A∧H, so thatU≤G. Further, indf bdA(U)=bdA(A∧ V)≤bdA(A)∨bdA(V)=0∨bdA(V)=bdA(V)≤A∧bd(V)≤bd(V). Thus bdA(U)≤ bd(V). Since indf bd(V)≤n−1, by induction hypothesis it follows that indf bdA(U)≤ n−1. Thus, for eachx∈Aand each open fuzzy setGinAsuch thatG(x)>0, there exists an open fuzzy setUinAsuch thatU(x)>0,U≤G, and indf bdA(U)≤n−1. Therefore byDefinition 3.5it follows that indfA≤n. Thus if indfX≤n, then indfA≤n. Therefore
the result holds forn. Hence indfA≤indfX.
Another new inductive dimension function for fuzzy topological spaces is introduced in the following.
Definition 3.7. LetX be an fts. The large inductive dimension ofX, denoted by IndfX, is defined as follows. IndfX= −1 if and only ifX=φ.IndfX≤n, for any nonnegative integern, if for each closed fuzzy setEand each open fuzzy setG inX such that E≤ G, there exists an open fuzzy set Uin Xsuch thatE≤U≤Gand Indf bd(U)≤n−1.
IndfX=nif IndfX≤nis true and IndfX≤n−1 is not true. IndfX= ∞if IndfX≤n is not true for everyn.
Note that ifXis a general topological space, then this concept reduces to that of Ind.
A relationship between indf and Indf is obtained in the following.
Theorem 3.8. IfXis an fts with the property that each open fuzzy set inXis union of closed fuzzy sets inX, then indfX≤IndfX.
Proof. This is proved by induction onn. Forn= −1, if IndfX≤ −1, then IndfX= −1, so thatX=φ. Therefore indfX= −1, so that indfX≤ −1. Thus if IndfX≤ −1, then indfX≤ −1. Therefore the result holds forn= −1. Assume that the result holds forn= k−1. That is, assume that if IndfX≤k−1, then indfX≤k−1. To prove that the result holds forn=k, suppose IndfX≤k. Then, to prove indfX≤k, letx∈X and letG be an open fuzzy set inX such thatG(x)>0. NowGis an open fuzzy set. By hypothesis, Gis union of closed fuzzy sets sayG= ∨Eλ, where eachEλ is a closed fuzzy set. Since G(x)>0, (∨Eλ)(x)>0, so that there exists aλ0 such thatEλ0(x)>0. AlsoEλ0≤Eλ≤G.
NowEλ0≤G, whereEλ0 is a closed fuzzy set andGis an open fuzzy set. Since IndfX≤ k byDefinition 3.7, there exists an open fuzzy set U in X such thatEλ0≤U≤G and Indf bd(U)≤k−1. By induction hypothesis, it follows that indf bd(U)≤k−1. Thus, for eachx∈Xand each open fuzzy setGsuch thatG(x)>0, there exists an open fuzzy setUinXsuch thatU(x)>0,U≤G, and indf bd(U)≤k−1. Therefore byDefinition 3.5 it follows that indfX≤k. Thus, if IndfX≤k, then indfX≤k. Therefore the result holds
forn=k. Hence indfX≤IndfX.
We also have the following result.
Theorem 3.9. IfXis an fts such that IndfX=0, thenXis a normal fts.
Proof. Leta,bbe closed fuzzy sets inXsuch thata≤1−b. Note that 1−bis an open fuzzy set. Since IndfX≤0, byDefinition 3.7, there exists an open fuzzy setcinX such
that a≤c≤1−b and Indf bd(c)≤0−1. That is, a≤c, c≤1−b, and Indf bd(c)≤
−1. Letd=1−c. Then a≤c,b≤d, c=1−d, and bd(c)=0. Since bd(c)=0, from Theorem 3.4, c is open, closed, and crisp. Thus, for each paira,bof closed fuzzy sets inX witha≤1−b, there exist open fuzzy setsc,dinX, such thata≤c,b≤d, andc≤1−d.
Therefore from [3, Theorem 5.2, page 36] it follows thatXis a normal fts.
A closed subset theorem for Indf is obtained, which is contained in the following.
Theorem 3.10. IfAis a closed crisp subspace of an ftsX, then IndfA≤IndfX.
Proof. This is proved by induction onn. Forn= −1, if IndfX≤ −1, then IndfX= −1, so thatX=φ. ThereforeA=φand so IndfA= −1, so that IndfA≤ −1. Thus if IndfX≤
−1, then IndfA≤ −1. Therefore the result is true forn= −1. Assume that the result holds forn=k−1. That is, assume that if IndfX≤k−1, then IndfA≤k−1. Then the result is to be proved forn=k, that is, to prove if IndfX≤k, then IndfA≤k. Suppose IndfX≤k. To prove IndfA≤k, letEbe a closed fuzzy set inA and letGbe an open fuzzy set inAsuch thatE≤G. SinceEis closed inAandAis closed inX, it follows that Eis closed inX. AlsoGis an open fuzzy set inA. ThereforeG=A∧H, whereH is an open fuzzy set inX. Also sinceE≤G, we haveE≤A∧H≤H, so thatE≤H, whereEis closed fuzzy set inXandH is open fuzzy set inX. Since IndfX≤k, by definition, there exists an open fuzzy setV inXsuch thatE≤V≤H and Indf bd(V)≤k−1. Therefore E∧A≤V∧A≤H∧Awhich impliesE≤U≤G, whereV∧A=Uis a closed fuzzy set inA. Also bdA(U) is a closed fuzzy set inA[7] andAis a closed fuzzy set inX. Therefore bdA(U) is a closed fuzzy set inX. Further bdA(U)≤bd(V). Therefore bdA(U) is closed in bd(V) and Indf bd(V)≤k−1. Therefore by induction hypothesis Indf bdA(U)≤k−1.
Thus for each closed fuzzy setEin A and open fuzzy setG in Asuch that E≤G, there exists an open fuzzy setU in A such thatE≤U≤G and Indf bdA(U)≤k−1.
Therefore, by definition, it follows that IndfA≤k. Thus, if IndfX≤k, then IndfA≤ k. Therefore the result holds forn=k. Thus the result holds for all values ofn. Hence
IndfA≤IndfX.
Acknowledgments
The authors are highly grateful to the University Grants Commission, New Delhi, and the Karnatak University, Dharwad, for their financial support to undertake this research work. The authors are also thankful to the reviewers and the Managing Editor for their comments which enabled to bring this paper to the present form.
References
[1] J. M. Aarts, A new dimension function, Proceedings of the American Mathematical Society 50 (1975), no. 1, 419–425.
[2] C. L. Chang, Fuzzy topological spaces, Journal of Mathematical Analysis and Applications 24 (1968), no. 1, 182–190.
[3] S. R. Malghan and S. S. Benchalli, On fuzzy topological spaces, Glasnik Matematiˇcki. Serija III 16(36) (1981), no. 2, 313–325.
[4] , Open maps, closed maps and local compactness in fuzzy topological spaces, Journal of Mathematical Analysis and Applications 99 (1984), no. 2, 338–349.
[5] A. R. Pears, Dimension Theory of General Spaces, Cambridge University Press, Cambridge, 1975.
[6] A. I. Va˘ınˇste˘ın, A class of infinite-dimensional spaces, Matematicheski˘ı Sbornik. Novaya Seriya 79 (121) (1969), 433–443.
[7] R. H. Warren, Boundary of a fuzzy set, Indiana University Mathematics Journal 26 (1977), no. 2, 191–197.
[8] M. D. Weiss, Fixed points, separation, and induced topologies for fuzzy sets, Journal of Mathemat- ical Analysis and Applications 50 (1975), no. 1, 142–150.
[9] L. A. Zadeh, Fuzzy sets, Information and Computation 8 (1965), 338–353.
S. S. Benchalli: Department of Mathematics, Karnatak University, Dharwad 580003, Karnataka, India
E-mail address:benchalli [email protected]
B. M. Ittanagi: Department of Mathematics, Karnatak University, Dharwad 580003, Karnataka, India
E-mail address:basuraj [email protected]
P. G. Patil: Department of Mathematics, Karnatak University, Dharwad 580003, Karnataka, India
E-mail address:pgpatil maths@rediffmail.com
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