© Associated Asia Research Foundation (AARF) A Monthly Double-Blind Peer Reviewed Refereed Open Access International e-Journal - Included in the International Serial Directories. Page | 1 S-REGULAR SPACES AND S-NORMAL SPACES IN TOPOLOGY Govindappa Navalagi Department of Mathematics, KIT Tiptur-572202, Karnataka, India. Sujata Mookanagoudar Department of Mathematics, Government First Grade College, Haliyal-581329, Karnataka, India. ABSTRACT The aim of this paper is to introduce and study some forms of weak regular spaces and weak forms of normal spaces , viz. s- regular spaces , s -regular spaces , s-normal spaces by using -closed sets and semiopen sets . Also , we studied some related functions like gs-closed functions , gs-continuous functions for preserving these regular spaces and normal spaces . Mathematics Subject Classification (2010) : 54 A 05 , 54 C 08 ; 54 D 15 Key words and Phrases : semiopen sets, -closed sets, -open sets, E.D. spaces ,PS-spaces , gs-closedness , s-continuity, -continuity, -irresoluteness, irresoluteness 1.Introduction In 1963, N.Levine [16] introduced and studied the concepts of semiopen sets and semi continuity in topological spaces and in 1983, M.E. Abd El–Monsef et al [1] have introduced the concepts of -open sets and -continuity in topology . Latter , these -open sets are recalled as semipreopen sets , which were introduced by D. Andrejevic in [5]. Further these semi open sets and -open sets ( = semipreopen sets ) have been studied by various authors in the literature see [2, 4,10,14,17,23]. In 1970 and 1990, respectively Levine [15] and S.P.Arya et al [6] have defined and studied the concept of g-closed sets and gs-closed sets in topology . The aim of this paper is to introduce and study some forms of weak regular spaces and weak forms of normal spaces , viz. s- regular spaces , s-regular spaces , s-normal spaces by using -closed sets and semiopen sets . Also , we studied some related functions like gs-closed functions , gs- continuous functions for preserving these regular spaces and normal spaces . International Research Journal of Mathematics, Engineering and IT ISSN: (2349-0322) Impact Factor- 5.489, Volume 5, Issue 9, September 2018 Website- www.aarf.asia, Email : , e
© Associated Asia Research Foundation (AARF) A Monthly Double-Blind Peer Reviewed Refereed Open Access International e-Journal - Included in the International Serial Directories. Page | 2 2. Preliminaries Throughout this paper X,Y will denote topological spaces on which no separation axioms assumed unless explicitly stated. Let f : X Y represent a single valued function. Let A be a subset of X. The closure and interior of A are respectively denoted by Cl(A) and Int(A). The following definitions and results are useful in the sequel. Definition 2.1: A subset A of X is called (i) semiopen (in short , s-open) set [16] if A ClInt(A) . (ii) preopen (in short , p-open) set [20] if A IntCl(A). (iii) -open [1] (=semipreopen [5]) set if A ClIntCl(A) . The complement of a s-open (resp. p-open , -open ) set is called s-closed[7] (resp. p-closed [13] , -closed [1] ) set. The family of s-open (resp. p-open , -open ) sets of X is denoted by SO(X) (resp. PO(X) , O(X)) . Definition 2.2 : The intersection of all s-closed (resp. -closed ) sets containing a subset A of space X is called the s-closure [7] (resp. -closure[ 2] ) of A and is denoted by sCl(A) (resp. Cl(A)). Definition 2.3 : The union of all s-open (resp. -open ) sets which are contained in A is called the s-interior[7] ( resp. the -interior [2]) of A and is denoted by sInt(A) (resp. Int(A)). Definition 2.4 [6]: A subset A of a space X is called gs-closed if sCl(A) U whenever A U and U is open set in X. Definition 2.5[ 22]: A space X is said to be -regular if for each closed set F and for each x X- F , there exist two disjoint -open sets U and V such that xU and F V. Definition 2.6 [19 ]: A space X is said to be s-regular if for each closed set F and for each x X- F , there exist two disjoint s-open sets U and V such that xU and F V. Definition 2.7 [12]: A space X is said to be semi-regular if for each semiclosed set F and for each x X-F , there exist two disjoint s-open sets U and V such that xU and F V. Definition 2.8 [18]: A space X is said to be s-normal if for any pair of disjoint closed subsets A and B of X , there exist disjoint s-open sets U and V such that AU and B V. Definition 2.9 [11]: A space X is said to be semi-normal if for any pair of disjoint semiclosed subsets A and B of X , there exist disjoint s-open sets U and V such that AU and B V. Definition 2.10 [25]: A space X is said to be submaximal if every dense set of X is open in X (i.e., every preopen set of X is open in X ).
© Associated Asia Research Foundation (AARF) A Monthly Double-Blind Peer Reviewed Refereed Open Access International e-Journal - Included in the International Serial Directories. Page | 3 Definition 2.11[14]: A space X is said to be extremely disconnected (in brief , E.D.,) space if Cl(G) is open set for each open set G of X . Definition 2.12[4]: PS- spaces .A space X is said to be s-regular [ ] if for each closed set F and for each x X-F , there exist two disjoint s-open sets U and V such that xU and F V. Definition 2.13 [17]: A function f: XY is said to be -irresolute if f-1(V) is is -open set in X for each for each -open set V in Y. Definition 2.14[8] : A function f: XY is said to be irresolute if f-1(V) is is s-open set in X for each for each s-open set V in Y. Definition 2.15[9] : A function f: XY is said to be presemiopen if f(V) is is semiopen set in Y for each for each semiopen set V in X. Definition 2.16[22] : A function f: XY is said to be M--closed if the image of each -closed set of X is -closed in Y. 3. Properties of s-regular spaces We, define the following. Definition 3.1 : A topological space X is said to be s-regular if for each -closed set F of X and each point x in X - F, there exist disjoint s-open sets U and V such that x U and F V. Clearly, every s-regular space is s-regular as well as semi-regular , since every closed set as well as s-closed set is -closed set. Definition 3.2 : A space X is said to be *-regular if for each -closed set F and for each x X- F , there exist -open sets U and V such that F U and xV. Theorem 3.3 : Every *-regular space is -regular. Proof : Let X be *-regular space and F be a closed set not containing x implies F be a - closed set not containing x. As X is *-regular space , there exist disjoint -open sets U and V such that x V and F U. Therefore X is -regular. Lemma 3.4: If A is subset of X and BO(X) such that AB= then Cl(A) B = . Proof : Let us assume that Cl(A) B implies there exists x such that x Cl(A) and x B. Now x Cl(A) A U for each -open set U containing x which is contradiction to hypothesis AB = for B O(X,x).Hence Cl(A) B = . Lemma 3.5 : Every s-regular space is *-regular space . Proof : Let X be s-regular space and F be a -closed set not containing x. As X is s-regular space , there exist disjoint s-open sets U and V such that x V and F U. But, every s-open set is -open set and thus X is *-regular.
© Associated Asia Research Foundation (AARF) A Monthly Double-Blind Peer Reviewed Refereed Open Access International e-Journal - Included in the International Serial Directories. Page | 4 Converse of Lemma-3.5 is not true . For example, Example 3.6 : Let X = { a,b,c}, = { X, , {a}, {b,c} } O(X) = { X, , {a}, {b}, {c}, {a,b},{a,c},{ b,c} } SO(X) = { X, , {a}, {b,c} } Clearly X is *-regular space but not s-regular . Since for -closed set {a,b} and c{a,b} there donot exist disjoint s-open sets U and V such that cU and F V. Theorem 3.7: For a topological space X the following statements are equivalent ; (a) X is s-regular (b) For each xX and for each -open set U containing x there exists a s-open set V containing x such that x V sCl(V)U. (c) For each -closed set F of X , {sCl(V)/FV and VSO(X)} = F (d) For each nonempty subset A of X and each U O(X) if A U then there exists VSO(X) such that AV and sCl(V) U (e) For each nonempty subset A of X and each FF(X) if A F = then there exists V,WSO(X) such that AV , FW and VW =. Proof : (a) (b) Let X be s- regular space. Let xX and U be -open set containing x implies X - U is -closed such that xX- U. Therefore by (a) there exists two s-open sets V and W such that xV and X-U W X-W U. Since VW = sCl(V) W = sCl(V) X- W U. Therefore , x V sCl(V) U. (b) (c) Let F be a -closed subset of X and xF , then X - F is -open set containing x. By (b) there exists s-open set U such that xU sCl(U) X-F implies F X-sCl(U) X-U i.e F V X-U where V = X-sCl(U) SO(X) and xV that implies x sCl(V) implies x { sCl(V) / FVSO(X)}. Hence , { sCl(V) / FVSO(X)} = F (c) (d) A be a subset of X and UO(X) such that A U . there exists x0 X such that. x0 A U. Therefore X-U is -closed set not containing x0 x0 Cl(X-U). By (c) , there exists W SO(X) such that X-U W x0 sCl(W). Put V = X- sCl(W) , then V is s-open set containing x0. A V and sCl(V) sCl(X-sCl(W)) sCl( X-W).Therefore , sCl(V) sCl(X-W) U. (d) (e) Let A be a nonempty subset of X and F be -closed set such that A F = .Then X-F is -open in X and A (X-F) . Therefore by (d) , there exist V SO(X) such that A V and sCl(V) X –F. Put W = X- sCl(V) then W SO(X) such that F W and WV =. (f) (a) Let x X be arbitrary and F be -closed set not containing x. Let A= X\F be a nonempty -open set containing x then by (e) ,there exist disjoint s-open sets V and W such that F W and A V xV. Thus X is a s-regular.
© Associated Asia Research Foundation (AARF) A Monthly Double-Blind Peer Reviewed Refereed Open Access International e-Journal - Included in the International Serial Directories. Page | 5 Theorem 3.8 : In a topological space X following statements are equivalent; (a) X is s-regular (b) for each -open set U of X containing x there exists s-open set V such that x V sCl(V) U Proof : (a) (b) Let x X and U be -open set of X containing x X - U is -closed set not containing x. As X is s- regular , there exist disjoint s-open sets V and W such that xV and X-U W X-W U. As VW = sCl (V) W = sCl(V) X-W U. Hence x V sCl(V) U. (b) (a) Let for each x X, F be -closed set not containing x , therefore X-F is -open set containing x hence from (b) there exists s-open set V such that x V sCl(V) X-F . Let U = X –sCl(V) then U is s-open set such that F U , x V and U V = . Thus there exists disjoint s-open sets U and V such that xV and F U. Therefore X is s- regular. We, define the following. Definition 3.9: A space X is said to be strongly *-regular if for each -closed set F and for each x X-F , there exists disjoint open sets U and V such that F U and xV. Theorem 3.10: Every strongly *- regular space is *-regular space. Proof : Let X be strongly *-regular space and F be any -closed set and xF. Then there exist disjoint open sets U and V such that xU and FV .Since every open set is -open and hence U and V are - open sets such that xU and F V. This shows that X is *-regular. Converse of the theorem 3.10 and other statements are not true in general. For, Example 3.11 : Let X = { a,b,c}, = { X, , {a}, {b}, {a,b} } O(X) = { X, , {a}, {b}, {a,b},{a,c},{ b,c} } Clearly X is *-regular but not strongly *-regular . Theorem 3.12 : For a topological space X the following statements are equivalent ; (a) X is strongly *-regular (b) For each xX and for each -open set U of X containing a point x , there exists an open set V such that xV Cl(V)U. (c) For each -closed set F of X , {Cl(V)/FV } = F . (d) For each nonempty subset A of X and each U O(X) if A U then there exists open set V such that AV and Cl(V) U . (e) For each nonempty subset A of X and each closed set F of X such that A F = , there exit , V,W O(X), such that AV , FW and VW =. The routine proof of the theorem is omitted.
© Associated Asia Research Foundation (AARF) A Monthly Double-Blind Peer Reviewed Refereed Open Access International e-Journal - Included in the International Serial Directories. Page | 6 Next , we prove some preservation theorems in the following. Theorem 3.13 : If f : X Y is a pre-semiopen , -irresolute bijection and X is s-regular space , then Y is s-regular . Proof : Let F be any -closed subset of Y and y Y with y F . Since f is -irresolute, f-1(F) is -closed set in X. Again, f is bijective, let f (x) = y , then x f-1(F).Since X is s- regular , there exist disjoint s-open sets U and V such that x U and f-1(F) V . Since f is , presemiopen bijection , we have y f(U) and F f(V) and f(U) f(V) = f (U V) = . Hence, Y is s-regular space. We , define the following. Definition 3.14 : A function f : X Y is said to be always--closed if the image of each - closed subset of X is -closed set in Y. Now, we prove the following. Theorem 3.15 : If f : X Y is an always -closed , irresolute injection and Y is s-regular space , then X is s-regular . Proof : Let F be any -closed set of X and x F. Since f is an always -closed injection, f(F) is -closed set in Y and f(x) f(F) . Since Y is s-regular space and so there exist disjoint s- open sets U and V in Y such that f(x) U and f(F) V . By hypothesis, f-1(U) and f-1(V) are s-open sets in X with x f-1(U) , F f-1(V) and f-1(U) f-1(V) = . Hence, X is s- regular space. We, define the following. Definition 3.16 : A function f : X Y is said to be s-continuous if the inverse image of each -open set of Y is s-open set in X. Next, we give the following. Theorem 3.17 : If f : X Y is an always -closed , s-continuous injection and Y is - regular space , then X is s-regular . 4.s-normal spaces. We , define the following.
© Associated Asia Research Foundation (AARF) A Monthly Double-Blind Peer Reviewed Refereed Open Access International e-Journal - Included in the International Serial Directories. Page | 7 Definition 4.1 : A function f : X Y is said to be gs-continuous if for each -closed set F of Y , f-1 (F) is gs-closed set in X. It is obvious that a function f : X Y is gs-continuous if and only if f-1(V) is gs-open in X for each -open set V of Y. Definition 4.2: A function f : X Y is said to be gs-closed if for each -closed set F of X , f(F) is gs-closed set in Y. We, recall the following . Definition 4.3[ 23] : A function f : X Y is said to be : (i)pre-gs-continuous if the inverse image of each s-closed set F of Y is gs-closed in X. (ii)pre-gs-closed if the image of each s-closed set of X is gs-closed in Y. Clearly, (i) every pre-gs-continuous function is pre-gs-continuous , (ii) every pre-gs-closed function is pre-gs-closed , since in both cases every s-closed set is -closed set. We ,prove the following. Theorem 4.4 : A surjective function f : XY is gs-closed if and only if for each subset B of Y and each -open set U of X containing f-1(B) , there exists a gs-open set V of Y such that B V and f-1(V) U. Proof : Suppose f is gs-closed function. Let B be any subset of Y and U be any -open set in X containing f-1(B) . Put V = Y – f(X-U). Then , V is gs-open set in Y such that B V and f-1(V) U. Conversely, let F be any -closed set of X . Put B = Y – f(F) , then we have f-1(B) X –F and X-F is -open in X. There exists a gs-open set V of Y such that B =Y-f(F) V and f-1(V) X –F .Therefore , we obtain that f(F) = Y-V and hence f(F) is gs-closed set in Y .This shows that f is gs-closed function. We,define the following. Definition 4.5 : A function f : XY is said to be strongly -closed if the image of each - closed set of X is closed in Y. Definition 4.5 : A function f : XY is said to be always gs-closed if the image of each gs- closed set of X is gs-closed in Y. Theorem 4.6: Let f : XY and g : Y Z be two functions . Then the composition function gof : X Z is gs-closed if f and g satisfy one of the following conditions :
© Associated Asia Research Foundation (AARF) A Monthly Double-Blind Peer Reviewed Refereed Open Access International e-Journal - Included in the International Serial Directories. Page | 8 (i) f is gs-closed and g is always gs-closed . (ii) f is strongly -closed and g is gs-closed . Proof : (i)Let H be any -closed set in X and f is gs-closed , then f(H) is gs-closed set in Y.Again , g is always gs-closed function and f(H) is gs-closed set in Y , then gof (H) is gs-closed set in Z . This shows that gof is gs-closed function. (ii)Let F be any -closed set in X and f is strongly -closed function, then f(F) is closed set in Y. Again, g is gs-closed function and f(F) is closed set in Y, then gof (F) is gs-closed set in Z. This shows that gof is gs-closed function. We, define the following. Definition 4.7 : A function f : X Y is said to be gs-closed if for each gs-closed set F of X , f(F) is -closed set in Y. Theorem 4.8 : Let f : XY and g : Y Z be two functions . Then , (i) if f is -losed and g is gs-closed , then gof is gs-closed . (ii) if f is gs-closed and g is strongly gs- closed , then gof is strongly -closed. (iii) if f is M--closed and g is gs-closed , then gof is gs-closed . (iv) if f is alays gs-closed and g is gs-closed , then gof is gs-closed. Proof : (i). Let H be a closed set in X , then f(H) be -closed set in Y since f is -closed function. Again , g is gs-closed and f(H) is -closed set in Y then g(f(H))= gof(H) is gs- closed set in Z .Thus, gof is gs-closed function. (ii)Let F be any -closed set in X and f is gs-closed function, then f(F) is gs-closed set in Y. Again , g is strongly gs-closed and f(F) is gs-closed set in Y , then gof(F) is closed set in Z .This shows that gof is strongly -closed function. (iii)Let H be any -closed set in X and f is M--closed function then f(H) is -closed set in Y. Again, g is gs-closed function and f(H) is -closed set in Y , then gof (H) is gs-closed set in Z. Therefore , gof is gs-closed function. (iv)Let H be any gs-closed set in X and f is always gs-closed function , then f(H) be gs-closed set in Y . Again , g is gs-closed function and f(H) is gs-closed set in Y, then gof (H) is -closed set in Z . This shows that gof is gs-closed function. Theorem 4.9 : Let f : XY and g : Y Z be two functions and let the composition function gof : X Z is gs-closed.Then, the following hold : (i) if f is -irresolute surjection , then g is gs-closed. (ii) if g is gs-irresolute injection , then f is gs-closed . Proof : (i) Let F be a -closed set in Y . Since f is -irresolute surjective , f-1(F) is -closed set in X and (gof)(f-1(B)) = g(F) is gs-closed set in Z . This shows that g is gs-closed function.
© Associated Asia Research Foundation (AARF) A Monthly Double-Blind Peer Reviewed Refereed Open Access International e-Journal - Included in the International Serial Directories. Page | 9 (ii)Let H be a -closed set in X .Then, gof (H) is gs-closed set in Z . Again , g is gs-irresolute injective , g-1(gof(H)) = f(H) is gs-closed set in Y . This shows that f is gs-closed function. We , define the following. Definition 4.10: A space X is said to be s-normal if for any pair of disjoint -closed sets A and B of X ,there exist disjoint s-open sets U and V such that A U and B V. Clearly, every s-normal space is s-normal as well as semi-normal , since every closed set as well as s-closed set is -closed set. Definition 4.11 : A space X is said to be *-normal if for any pair of disjoint -closed sets A and B of X ,there exist disjoint open sets U and V such that A U and B V. Clearly , every *-normal space is s-normal space. We , recall the following. Lemma 4.12 [6] : A subset A of a space X is gs-open iff F sInt(A) whenever F A and F is closed set in X. Lemma 4.13 [3] : If X is submaximal and E.D. space , then every -open set in X is open set. We, characterize the s-normal spaces in the following. Theorem 4.14 : The following statements are equivalent for a submaximal and E.D., space X : (i) X is s-normal space, (ii) For any pair of disjoint -closed sets A , B of X , there exist disjoint gs-open sets U , V such that A U and B V, (iii) For any -closed set A and any -open set V containing A , there exists a gs-open set U such that A U sCl(U) V. Proof : (i) (ii) . Obvious , since every s-open set is gs-open set. (ii)(iii) . Let A be any -closed set and V be any -open set containing A . Since A and X-V are disjoint -closed sets of X , there exist gs-open sets U and W of X such that A U and X-V W and U W = . By Lemmas – 4.12 and 4.13 , we have X –V sInt (W) . Since U sInt(W) = , we have sCl(U) sInt(W) = and hence sCl(U) X-sInt(W) V. Thus , we obtain that A U sCl(U) V . (iii) (i). Let A and B be any disjoint -closed sets of X . Since X-B is -open aet containing A, there exists a gs-open set G such that A G sCl(G) X-B. Then by Lemmas-4.12 and 13 , A sInt(G) . Put U = sInt (G) and V = x-sCl(G). Then , U and V are disjoint s-open sets such that A U and B V . Therefore, X is s-normal space. Theorem 4.15 : If f : X Y is a -irresolute , gs-closed surjection and X is s-normal, then Y is s-normal space.
© Associated Asia Research Foundation (AARF) A Monthly Double-Blind Peer Reviewed Refereed Open Access International e-Journal - Included in the International Serial Directories. Page | 10 Proof : Let A and B be any disjoint -closed sets of Y. Then, f-1(A) and f-1(B) are disjoint - closed sets of X since f is -irresolute function . Since X is s-normal , there exist disjoint s-open sets U and V in X such that f-1(A) U and f-1(B) V . By Th.4.4 , there exist gs-open sets G and H of Y such that A G , B H , f-1(G) U and f-1(H) V . Then , we have f-1(G) f-1(H) = and hence G H = . It follows from Th.4.14 that space Y is s-normal. References [1] M.E.Abd El-Monsef , S.N.El-Deeb and R.A.Mahamoud, -open sets and -continuous mappings, Bull.Fac.Sci. Assiut Univ. , 12(1983) ,77-90. [2] M.E.Abd El-Monsef , R.A.Mahamoud and E.R. Lashin, -closure and -interior, J.Fac.Ed. Ain Shams Univ., 10(1986),235-245. [3] M.E. Abd El-Mosef and A.M.Kozae,On extremally dixconnectedness, ro-equivalence and properties of some maximal topologies, 4th Conf.Oper.Res.Math.Methods,(Alex.Univ.,1988). [4] T.Aho and T.Nieminen, Spaces in which preopen sets are semiopen , Ricerche Mat., (1994) (to appear). [5] D.Andrijevic, Semi-preopen sets , Mat.Vesnik , 38(1986) , 24 - 32. [6] S.P.Arya and T.M.Nour , Characterizations of s-normal spaces, Indian J.Pure Appl.Math., 21(8) ,(1990) , 717-719. [7] S.G.Crossley and S.K.Hildebrand , Semiclosure, Texas J.Sci., 22 (1971) , 99-112. [8] S.G.Crossley and S.K.Hildebrand , Semiclosed sets and semicontinuity in topological spaces , Texas J.Sci., 22(1971) , 125-126. [9] S.G.Crossley and S.K.Hildebrand , Semitopological properties, Funda .Math., 74 (1972) , 233-254. [10] J.Dontchev , The characterizations of spaces and maps via semipreopen sets, Indian J.Pure And Appl.Math., 25(9)(1994), 939-947. [11] C.Dorcett , Semi-normal spaces , Kyungpook Math. J., 25(1985),173-180. [12] C.Dorsett , Semi-regular spaces, Soochow J. Math., 8 (1982),45-53. [13] S. N. El- Deeb, I. A. Hasanien, A.S. Mashhour and T. Noiri, On p-regular spaces,
© Associated Asia Research Foundation (AARF) A Monthly Double-Blind Peer Reviewed Refereed Open Access International e-Journal - Included in the International Serial Directories. Page | 11 Bull. Math. Soc. Sci. Math. R. S. R. 27 (75) (1983), 311-315. [14] L.Gillman and M.Jerison , Rings of continuous functions ,Univ.Series of Higher Math., Van Nostrand ,Princeton,New York ,(1960). [15] N.Levine , Generalized closed sets in topology , Rend.Circ.Mat. Palermo , (2) 19 (1970), 89-96. [16] N.Levine, Semi-open sets and semi continuity in topological spaces, Amer.Math. Monthly , 70 (1963) , 36-41. [17] R.A.Mahamoud and M.E.Abd El-Monsef, -irresolute and - topological invariant, Proc. Pakistan Acad. Sci., 27(1990), 285—296. [18] S.N.Maheshwari and R.Prasad,On s-normal spaces,Bull.Math.Soc.Sci.R.S.Roumanie, 22(70) ,(1978),27-29. [19] S.N.Maheshwari and R.Prasad,On s-regular spaces,Glasnik Mat. , 30(10) (1975),347-350. [20] A.S.Mashhour,M.E.Abd El-Monsef and S.N.El-Deeb, On Precontinuous and Weak Pre- continuous Mappings. Proc. Math. Phys.Soc. Egypt,53(1982),47-53. [21] G.B.Navalagi , On semipre-continuous functions and properties of generalized semi-preclosed sets in topology, I J M S , 29(2) (2002) , 85-98. [22] T.Noiri , Weak and strong forms of - irresolute functions., Acta.Math.Hungar. , 99(4) (2003), 315-328. [23] T.Noiri , On s-normal spaces and pre-gs-closed functions , Acta Math., Hungar.80(1-2) (1998) , 105-113. [24] J.H.Park and Y.B.Park, On sp-regular spaces, J.Indian Acad.Math. 17(1995) ,No.2 , 212-218. [25] I.L.Reilly and M.K.Vamanmurthy, On some questions concerning preopen sets, Kyungpook Math.J., 30(1990), 87-93. |
