g -Pre Regular and g -Pre Normal Spaces

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1 International Mathematical Forum, 4, 2009, no. 48, g -Pre Regular and g -Pre Normal Spaces S. S. Benchalli Department of Mathematics Karnatak University, Dharwad Karnataka State, India. benchalli math@yahoo.com T. D. Rayanagoudar and P. G. Patil Department of Mathematics SKSVM Agadi College of Engg. and Techn., Laxmeshwar , Karnataka State, India rgoudar math@yahoo.com pgpatil01@gmail.com Abstract The aim of this paper is to introduce and study two new classes of spaces, called g -pre regular and g -pre normal spaces. Some basic properties of these separation axioms are studied by utilizing g -preclosed and g -preopen sets. A subset A of a space X is called g -preclosed (briefly g p-closed) set if pcl(a) U whenever A U and U is g-open in X. Mathematics Subject Classification: 54D10, 54D15, 54C08, 54C10 Keywords: g p-closed set, g p-open set, g p-regular spaces, g p-normal spaces, g p-continuity 1 Introduction The notion of pre-open set [13] plays a significant role in general topology. The most important generalizations of regularity (resp. normality) are the notions of pre-regularity [4] and strong regularity (resp. pre-normality [15], strong normality [15]). Levine [8] in 1963, started the study of generalized open sets with the introduction of semi-open sets. Then Njatsad [18] studied α-open sets; In 1982, Mashhour et. al [13] introduced preopen sets and pre-continuity in topology.

2 2400 S. S. Benchalli, T. D. Rayanagoudar and P. G. Patil Since then many topologists have utilized these concepts to the various notions of subsets, weak separation axioms, weak regularity, weak normality and weaker and stronger forms of covering axioms in the literature. The concept of s-regular and s-normal spaces in topological spaces were introduced and studied by Maheshwari and Prasad [10, 11]. Arya and Nour [2] obtained some characterization of s-normal spaces. Munshi [16] introduced and studied the notions of g-regular and g-normal spaces using g-closed sets in topological spaces. Further Noiri and Popa [19] investigated the concepts introduced by Munshi [16]. In 2002, Veerakumar [21] have defined the notion of g p-closed sets, g p- continuity and g p-irresolute maps. In this paper we utilize these sets to define and study the two new classes of spaces, called g -pre regular and g -pre normal spaces in topology. Also we characterize their basic properties along with already existing weaker forms of regularity and normality. 2 Preliminaries Throughout this paper (X, τ), (Y,σ) and (Z, η) represent non-empty topological spaces on which no separation axioms are assumed unless explicitly stated and they are simply written X, Y and Z respectively. For a subset A of (X, τ), the closure of A, the interior of A with respect to τ are denoted by cl(a) and int(a) respectively. The complement of A is denoted by A c. Before entering into our work we recall the following definitions from various authors. Definition 2.1. A subset A of a topological space (X, τ) is called semiopen [8] (resp. pre-open [13] and α-open [14]) set if A cl(int(a)) (resp. A int(cl(a)) and A int(cl(int(a)))). The complement of semi-open (resp. pre-open and α-open) set is called semi-closed (resp. pre-closed and α-closed) set. Definition 2.2. A subset A of a topological space (X, τ) is called 1. g-closed [9] if cl(a) G whenever A G and G is open in (X, τ). The complement of g-closed set is called g-open. 2. g p-closed [21] if pcl(a) G whenever A G and G is g-open in (X, τ). The complement of g p-closed set is called g p-open. Definition 2.3. [6, 13] Let A X. 1. The intersection of all preclosed sets containing A is called preclosure of A and is denoted by pcl(a).

3 g -Pre Regular and g -Pre Normal Spaces The union of preopen sets contained in A is called preinterior of A and is denoted by pint(a). Definition 2.4. A topological space (X, τ) is said to be 1. strongly regular [15] if for each preclosed set A and each point x/ A, there exist disjoint preopen sets U and V in X such that A U and x V. 2. strongly normal [15] if for each pair of disjoint preclosed sets A and B of X, there exist disjoint preopen sets U and V containing them. 3. g-regular [16] if for each g-closed set F and each point x/ F, there exist disjoint open sets U and V in X such that F U and x V. 4. g- normal [16] if for each pair of disjoint g-closed sets A and B of X, there exist disjoint open sets U and V such that A U and B V. Theorem 2.5. [13] If U is open and A is preopen then U A is preopen. Definition 2.6. [4] A subset A of a space X is said to be pre-regular if A is both preopen and preclosed. Definition 2.7. [21] A subset A of a topological space (X, τ) is called a αtp -space if every g p-closed set is preclosed. Definition 2.8. A map f : X Y is called 1. pre-continuous [13] if f 1 (V ) is pre-open in X for every open set V in Y. 2. g-continuous [3] if f 1 (F )isg-closed in X for every closed set F in Y. 3. M-preopen (resp. M-preclosed) [15] if f(v ) is preopen (resp. preclosed) set in Y for every preopen (resp. preclosed) set V of X. 4. g p-continuous [21] if f 1 (F )isg p-closed in X for every closed set F in Y. 5. gc-irresolute [12] if f 1 (F )isg-closed in X for every g-closed set F in Y. 6. g p-irresolute [21] if f 1 (F )isg p-closed in X for every g p-closed set F in Y. 7. pre-irresolute [20] if f 1 (F ) is pre-open in X for every pre-open set F in Y. 8. g p-closed map [21] if f(f )isg p-closed in Y for every closed set F in X. Lemma 2.9. [6] If X 0 α(x) and A PO(X), then X 0 A PO(X 0 ).

4 2402 S. S. Benchalli, T. D. Rayanagoudar and P. G. Patil 3 g -Pre Regular Spaces In this section, we introduce g -pre regular spaces in topological spaces. We obtain several characterizations of g -pre regular spaces. Definition 3.1. Aspace(X, τ) is said to be g -pre regular (briefly g p- regular) if for every g p-closed set F and a point x/ F, there exist disjoint pre-open sets U and V such that F U and x V. Clearly, every g p-regular space is strongly-regular space, but not conversely. Theorem 3.2. For a topological space (X, τ), the following are equivalent: (i) (X, τ) is g p-regular. (ii) Every g p-open set U is a union of pre-regular sets. (iii) Every g p-closed set A is an intersection of pre-regular sets. Proof. (i) (ii): Let U be a g p-open set and let x U. IfA = X U, then A is g*p-closed. By assumption there exist disjoint open subsets W 1 and W 2 of X such that x W 1 and A W 2.IfV = pcl(w 1 ), then V is pre-closed and V A V W 2 = φ. It follows that x V U. Thus U is a union of pre-regular sets. (ii) (iii): This is obvious. (iii) (i): Let A be g p-closed and let x / A. By assumption, there exists a pre-regular set V such that A V and x/ V.IfU = X\V, then U is a preopen set containing x and U V = φ. Thus(X, τ) isg p-regular. g p-open sets give rise to various separation properties of which we offer the following Definition 3.3. A topological space (X, τ) is called a (i) g p-t 0 -space if for each pair of distinct points there exists an g p-open set containing one point but not the other. (ii) (p, g p)-r 0 -space if pcl({x}) U whenever U is g p-open and x U. Definition 3.4. [7] A space (X, τ) is said to be pre-t 2 if for each pair of distinct points x and y in X, there exist disjoint preopen sets U and V in X such that x U and y V. Theorem 3.5. Every g p-regular space (X, τ) is both pre-t 2 and (p, g p)- R 0. Proof. Let (X, τ) beg p-regular space and let x, y X such that x/ y. By Theorem 3.2, {x} is either preopen or preclosed since every space is pre-t 2. If {x} is preopen, hence g p-open, then pre-regular by Theorem 3.2. Thus {x} and X {x} are separating preopen sets. If {x} is preclosed, then X {x} is

5 g -Pre Regular and g -Pre Normal Spaces 2403 preopen and so, by Theorem 3.2, the union of pre-regular sets. Hence there is a pre-regular set V X {x} containing y. This proves that (X, τ) is pre-t 2. By Theorem 3.2, it follows immediately that (X, τ) is also (p, g p)-r 0. Again we define the following. Definition 3.6. The intersection of all g p-closed sets containing A is called g -preclosure (g p-closure) of A and is denoted by g pcl(a). We now give the charecterizations of g pcl(a) in the following Theorem 3.7. Let A be a subset of a space X an x X. The following properties hold for the g pcl(a): 1. x g pcl(a) if and only if A U φ, for every U PO(X) containing x. 2. A is g p-closed if and only if A = g pcl(a). 3. g pcl(a) is g p-closed. 4. g pcl(a) g pcl(b) if A B. 5. g pcl(g pcl(a)) = g pcl(a). Proof. The proof is obvious. Definition 3.8. A subset N of X is called a g -pre-neighbourhood (g p- neighbourhood) of a point x in X if there exists a g p-open set U such that x U N. Theorem 3.9. Suppose that B A X, B is a g p-closed relative to A and that A is open and g p-closed in (X, τ). Then B is g p-closed in (X, τ). Theorem If (X, τ) is a g p-regular space and Y is an open and g p-closed subset of (X, τ), then the subspace Y is g p-regular. Proof. Let F be any g p-closed subset of Y and y F c. By Theorem 3.9, F is g p-closed in (X, τ). Since (X, τ) isg p-regular, there exist disjoint pre-open sets U and V of (X, τ) such that y U and F V. By Lemma 2.9 and also Y is open and hence α-open, we get U Y and V Y are disjoint pre-open sets of the subspace Y such that y U Y and F V Y. Hence the subspace Y is g p-regular.

6 2404 S. S. Benchalli, T. D. Rayanagoudar and P. G. Patil Theorem Let (X, τ) be a topological space. Then the following statements are equivalent: (i) (X, τ) is g p-regular. (ii) For each point x X and for each g p-open neighbourhood W of x, there exists a pre open set U of x such that pcl(u) W. (iii) For each point x X and for each g p-closed set F not containing x, there exists a pre-open set V of x such that pcl(v ) F = φ. Proof. (i) (ii): Let W be any g p-open neighbourhood of x. Then there exists an g p-open set G such that x G W. Since G c is g p-closed and x / G c, by hypothesis, there exist pre-open sets U and V such that G c U, x V and U V = φ and so V U c. Now pcl(v ) pcl(u c )=U c and G c U implies U c G W. Therefore pcl(u) W. (ii) (i): Let F be any g p-closed set and x/ F. Then x F c and F c is g p-open and so F c is an g p-neighbourhood of x. By hypothesis, there exists a pre-open set V of x such that x V and pcl(v ) F c, which implies F (pcl(v )) c. Then (pcl(v )) c is pre-open set containing F and V (pcl(v )) c = φ. Therefore X is g p-regular. (ii) (iii): Let x X and F be a g p-closed set such that x/ F. Then F c is an g p-neighbourhood of x and by hypothesis, there exists a pre-open set V of x such that pcl(v ) F c and hence pcl(v ) F = φ. (iii) (ii): Let x X and W be a g p-neighbourhood of x. Then there exists a g p-open set G such that x G W. Since G c is g p-closed and x / G c, by hypothesis, there exists a pre-open set U of x such that pcl(u) G c = φ. Therefore pcl(u) G W. Theorem A topological space (X, τ) is g p-regular if and only if for each g p-closed set F of (X, τ) and each x F c, there exist pre-open sets U and V of (X, τ) such that x U and F V and pcl(u) pcl(v )=φ. Proof. Let F be a g p-closed set of (X, τ) and x/ F. Then there exist pre-open sets U x and V such that x U x, F V and U x V = φ. Which implies that U x pcl(v )=φ. Since (X, τ) isg p-regular, there exist pre-open sets G and H of (X, τ) such that x G, pcl(v ) H and G H = φ. This implies pcl(g) H = φ. Now put U = U x G, then U and V are pre-open sets of (X, τ) such that x U and F V and pcl(u) pcl(v )=φ. Converse is obvious. Theorem If (X, τ) is g p-regular space and f :(X, τ) (Y,σ) is bijective, g p-irresolute and M-pre-open, then (Y,σ) is g p-regular. Proof. Let y Y and F be any g p-closed subset of (Y,σ) with y/ F. Since f is g p-irresolute, f 1 (F )isg p-closed set in (X, τ). Since f is bijective, let f(x) =y, then x/ f 1 (y). By hypothesis, there exist pre-open sets U and

7 g -Pre Regular and g -Pre Normal Spaces 2405 V such that x U and f 1 (F ) V. Since f is M-pre-open and bijective, we have y f(u) and F f(v ) and f(u) f(v )=f(u V )=φ. Hence (Y,σ) is g p-regular space. Theorem If f :(X, τ) (Y,σ) is gc-irresolute, M-pre-closed and A is a g p-closed subset of (X, τ), then f(a) is g p-closed. Theorem If f :(X, τ) (Y,σ) is gc-irresolute, M-pre-closed and injective and (Y,σ) is g p-regular, then (X, τ) is g p-regular. Proof. Let F be any g p-closed set of (X, τ) and x/ F. Since f is gcirresolute, M-pre-closed, by Theorem 3.14, f(f )isg p-closed in Y and f(x) / f(f ). Since (Y,σ) isg p-regular and so there exist disjoint pre-open sets U and V in (Y,σ) such that f(x) U and f(f ) V. By hypothesis, f 1 (U) and f 1 (V ) PO(X) such that x f 1 (U), F f 1 (V ) and f 1 (U) f 1 (V )= φ. Therefore (X, τ) isg p-regular. 4 g -Pre Normal Spaces In this section, we introduce and study the weak form of normality, called g -prenormality in topological spaces. Definition 4.1. A topological space (X, τ) is said to be g -pre-normal (g p-normal) if for any pair of disjoint g p-closed sets A and B, there exist disjoint pre-open sets U and V such that A U and B V. Clearly every strongly nomral space is g p-normal, but converse is not true in general. Theorem 4.2. If (X, τ) is a g p-nomral space and Y is an open and g p- closed subset of (X, τ), then the subspace Y is g p-normal. Proof. Let A and B be any two disjoint g p-closed sets of Y. By Theorem 3.9, A and B are g p-closed in (X, τ). Since (X, τ) isg p-normal, there exist disjoint pre-open sets U and V of (X, τ) such that A U and B V. Since Y is open and hence α-open. By Lemma 2.9, U Y and V Y are disjoint pre-open sets of the subspace Y. Hence the subspace Y is g p-normal. Now we charecterize the g p-normal spaces. Theorem 4.3. Let (X, τ) be a topological space. Then the following statements are equivalent. (i) (X, τ) is g p-normal. (ii) For each g p-closed F and for each g p-open set U containing F, there exists a pre-open set V containing F such that pcl(v ) U.

8 2406 S. S. Benchalli, T. D. Rayanagoudar and P. G. Patil (iii) For each pair of disjoint g p-closed sets A and B in (X, τ), there exists a pre-open set U containing A such that pcl(u) B = φ. (iv) For each pair of disjoint g p-closed sets A and B in (X, τ), there exist a pre-open sets U and V such that A U, B V and pcl(a) pcl(b) =φ. Proof. (i) (ii): Let F be a g p-closed set and U be a g p-open set such that F U. Then F U c = φ. By assumption, there exist pre-open sets V and W such that F V, U c W and V W = φ, which implies pcl(v ) W = φ. Now, pcl(v ) U c pcl(v ) W = φ and so pcl(v ) U. (ii) (iii): Let A and B be disjoint g p-closed sets of (X, τ). Since A B = φ, A B c and B c is g p-open. By assumption, there exists pre-open set U containing A such that pcl(u) B c and so pcl(u) B = φ. (iii) (iv): Let A and B be any g p-closed sets of (X, τ). Then by assumption, there exists a pre-open set U containing A such that pcl(u) B = φ. Since pcl(a) is pre-closed, it is g p-closed and so B and plc(a) are disjoint g p-closed sets in (X, τ). Therefore again by assumption, there exists a preopen set V containing B such that pcl(a) pcl(b) = φ. (iv) (i): Let A and B be any disjoint g p-closed sets of (X, τ). By assumption, there exist pre-open sets U and V such that A U, B V and pcl(u) pcl(v )=φ, we have U V = φ and thus (X, τ) isg p-normal. Theorem 4.4. If f :(X, τ) (Y,σ) is g p-irresolute, bijective, M-preopen mapping and (X, τ) is g p-normal, then (Y,σ) is g p-normal. Proof. Let A and B be any two disjoint g p-closed sets of (Y,σ). Since the map f is g p-irresolute, f 1 (A) and f 1 (B) are disjoint g p-closed sets of (X, τ). As (X, τ) isg p-normal, there exist disjoint pre-open sets U and V such that f 1 (A) U and f 1 (B) V. Since f is M-pre-open and bijective, we have f(u) and f(v ) are pre-open sets in (Y,σ) such that A f(u) and B f(v ) and f(u) f(v )=φ. Therefore (Y,σ) isg p-normal. Theorem 4.5. If f :(X, τ) (Y,σ) is gc-irresolute, M-pre-closed, preirresolute injection and (Y,σ) is g p-normal, then (X, τ) is g p-normal. Proof. Let A and B be any two disjoint g p-closed sets of (X, τ). Since f is gc-irresolute and M-pre-closed, f(a) and f(b) are disjoint g p-closed sets of (Y,σ) by Theorem Since (Y,σ) isg p-normal, there exist disjoint preopen sets U and V such that f(a) U and f(b) V. That is A f 1 (U), B f 1 (V ) and f 1 (U) f 1 (V )=φ. Since f is pre-irresolute, f 1 (U) and f 1 (V ) are pre-open sets in (X, τ), we have (X, τ) isg p-normal. References [1] D. Andrijevic, Semi-preopen sets, Mat Versnik, 38 (1) (1986),

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