The International journal of analytical and experimental modal analysis

ISSN NO: 0886-9367

Q** - Closed sets in topological spaces Dr.P.Padma#1 and C.Deepadevi#2 #1

#2

Assistant Professor, PG Student, Department of Mathematics, Idhaya College For Women, Kumbakonam. Tamilnadu, India. 1

2

[email protected] [email protected]

Abstract— In the year 2010, the concepts of Q*- closed sets were introduced and studied by Murugalingam and Lalitha [7, 8] in topological spaces. In this paper we introduce and study new type of closed sets called Q** - closed sets in topological spaces. Also we discuss some of their properties. Keywords— Q** - closedset,Q** - open set I. INTRODUCTION The generalized closed set has been first studied and initiated by N.Levine in the year 1970 [6]. This generalized closed set has lead to significant contribution to generalization of continuity. Bhattacharya and Lahiri [1] introduced a closed set namely sg closed set. Further Elvina Mary L and R.Myvizhi [3] has worked on gs* - closed set. In the year 2010, the concepts of Q*closed sets were introduced and studied by Murugalingam and Lalitha [7, 8] in topological spaces. It was observed that every Q* - closed set is a closed set. They have also discussed the Q*-continuous and Q*- irresolute functions in topological spaces. Further K.Kannan and K. Chandrasekhara Rao [5] extended the concepts of Q* - closed sets to topological spaces in 2013. In this paper we introduce and study new type of closed sets called Q** - closed sets in topological spaces. Also we discuss some of their properties in topological spaces. II. PRELIMINARIES Let (X, τ) or simply X denotes a topological space. For any subset A ⊆ X, the interior of A is the largest open set contained in A and the closure of A is the smallest closed set containing A and they are denoted by int (A) and cl (A) respectively. Now we shall require the following known definitions are prerequisites Definition 2.1 [7, 8] A subset A of a topological space (X, τ) is called i) a Q* - closed if int (A) = φ and A is closed. ii) a Q* - open if cl (A) = X and A is open. Definition 2.2 Let (X, τ) be a topological space. The set of all Q* - closed sets with X is a topology. It is denoted by τQ*. Theorem 2.1 Let (X, τ) be a topological space. The set of all Q* - closed sets with X is a topology. Definition 2.3 Let (X, τ) be a topological space. Let A ⊂ X. The intersection of all Q* - closed sets containing A is called the Q* closure of A and it is denoted by Q*cl (A) or τi - Q*cl (A). Definition 2.4 Let (X, τ)be a topological space. Let A ⊂X.The union of all Q* - open sets containing A is called theQ*- interior of A and it is denoted by Q* int (A) or τi - Q* int (A). Definition 2.5 [11] A subset A of a space X is said to be nowhere dense if int (cl (A)) = φ, Definition 2.6 [10] A topological space (X, τ) is called hyperconnected if the intersection of two non – empty open sets is non-empty. III. Q** - CLOSED SET In this section, we introduce and study new type of closed sets called Q** - closed sets in topological spaces. Definition 3.1 A subset A of a topological space (X, τ) is called Q**- closed if int (A) =φ and A is Q** closed.

Volume XII, Issue I, January/2020

Page No:69

The International journal of analytical and experimental modal analysis

ISSN NO: 0886-9367

Example 3.1 X= {a, b, c} and τ= {φ, x, {b}} Thenφ, {a, c} are Q** closed. Remark 3.1 It is obvious that every Q** closed set is closed, but the converse is not true is general. The following example supports one claim. Example 3.2 X= {a, b, c} and τ= {φ, x, {a}, {b, c}}Here {b, c} is closed, but not Q**- closed. Theorem 3.1 If A and B are Q**-closed sets then so is A∪B. Proof: Suppose that A and B are Q** -closed sets Then int (A) = int (B) =φ and A, B are Q*-closed. Consequently, A∪Bis Q**- closed Now, AC∩ int (A∪B) ⊆ int (A∪B)Let x∈ AC ∩ int (A∪B). Then x∈AC and x∈int (A∪B).Therefore, x∈B.Consequently,AC ∩ int (A∪B) ⊆B. But int (B) =φ.Hence AC ∩ int (A∪B) =φ. But int (A) =φ.This implies that int (A∪B) =φ.It follows that,A∪B is Q**-closed. Theorem 3.2 Every Q**-closed set is δ-set. Proof: Let A be Q**-closed.Then int (A) =φ and A is Q*-closed. Consequently,int [cl [int (A)]] =int [cl (φ)] =int (φ) =φTherefore, A is δ-set. Theorem 3.3 If A is Q** -closed then A is nowhere dense. Proof: Since A is Q*-closed, we have int (A) =φ and A is Q**-closed.Therefore, cl [int (A)] =φ.Hence A is nowhere dense. Definition 3.2. A set A of a topological space (X, τ) is called Q**-open it its complement AC is Q**-closed in X. Example 3.3. In example 3.1, X and {b} are Q**- open. Theorem 3.4. A set A of a topological space (X, τ) is Q**- open if and only if cl (A) =X and A is open. Proof. Necessity: Suppose that A is Q**- open. ThenAC is Q** - closed. Therefore, int (AC) = [cl (A)] C=φ and AC is closed. Consequently,cl (A) =X and A is open. Sufficiency: Suppose that cl (A) =X and A is open. Then[cl (A)] C= int (AC) =φ and A and ACis closed. Consequently, C A is Q**- closed. This completes the proof. Corollary 3.1. A set A of a topological space (X, τ) is Q**- open if and only if A is dense and open. Theorem 3.5. X is not -Q** closed;Xis -Q** open;Φ is not - Q** open; Φ is - Q** closed. Remark 3.3 It is obvious that every Q**- open set is open, but the converse is not true in general. The following example supports our claim. Example 3.4. In example 3.3, {b, c} is open, but not Q**- open.

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The International journal of analytical and experimental modal analysis

ISSN NO: 0886-9367

IV. CONCLUSION In this paper, the new type of closed sets called Q** - closed sets in topological spaces are introduced and studied. Also we discuss some of their properties. REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]

P.Bhattacharya, and B.K.Lahiri, Semi generalized closed sets in topology, Indian J.Math, 29(3) 1987, 375 – 382. N. Bourbaki, General topology, Part I, Addison Wesley, Reading, Mass., 1966. Elvina Mary L and R.Myvizhi, gs* - closed sets in topological spaces, International Journal of Math Trends and Technology, 72(2014). R.James Munkres, Topology, A First -Course, Prentice Hall of India, New Delhi, (2001). K.Kannan and K.Chandrasekhararao, τ1τ2 - Q* closed sets, Thai Journal of Mathematics Volume 11 (2013) Number 2: 439 - 445. N.Levine, Generalized closed sets in topology, Rend. Circ. Mat. Palermo, 19 (2) (1970), 89-96. M. Murugalingam and N. Laliltha , Q star sets , Bulletin of pure and applied Sciences , Volume 29E Issue 2 ( 2010 ) p. 369 - 376 . M. Murugalingam and N. Laliltha, Q* sets in various spaces, Bulletin of pure and applied Sciences, Volume 3E Issue 2 (2011) p. 267 - 277. P.Padma, New separation axioms, new type of compact spaces and new class of functions in bitopological spaces, September 2018, Ph.D. thesis. L.A.Steen and J.A.Seeback Jr., Counter examples in topology, (Hold, Rinerhart and Winston. Inc., New York-Montreal-London, 1970). S. Willard, General Topology, Addison-Wesley Publishing Company, Reading, Massachusetts, (1970).

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The International journal of analytical and experimental modal analysis

Volume XII, Issue I, January/2020

ISSN NO: 0886-9367

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