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ISSN 2320-5407 International Journal of Advanced Research (2013), Volume 1, Issue 10, 539-545 Journal homepage: http://www.journalijar.com INTERNATIONAL JOURNAL OF ADVANCED RESEARCH RESEARCH ARTICLE Fine ππΉπ βSeparation Axioms in Fine β Topological Sapce K. Rajak Assistant Professor, Lakshmi Narain College of Technology, Jabalpur Manuscript Info Abstract Manuscript History: In this paper, we have introduce, study and investigated the following separation axioms: π β ππΏπ β ππ space (i=0, 1, 2) and weaker forms of regular and normal space by using the notion of π β ππΏπ βopen sets. Received: 10 November 2013 Final Accepted: 28 November 2013 Published Online: December 2013 Key words: Fine-semi-open set, π β ππΏπ βclosed set, π β ππΏπ β π0 βspace, π β ππΏπ β π1 βspace, π β ππΏπ β π2 βspace, π β ππΏπ βregular. Copy Right, IJAR, 2013,. All rights reserved. 1. Introduction Powar P. L. and Rajak K. [16], have investigated a special case of generalized topological space called fine topological space. In this space, they have defined a new class of open sets namely fine-open sets which contains all πΌ βopen sets, π½ βopen sets, semi-open sets, pre-open sets, regular open sets etc.. By using these fine-open sets they have defined fine-irresolute mappings which include pre-continuous functions, semi-continuous function, πΌ βcontinuouos function, π½ βcontinuous functions, πΌ βirresolute functions, π½ βirresolute functions, etc (cf. [12][16]). In this paper, we have defined strongly ππΏπ βcontinuous functions, pre-ππΏπ βcontinuous functions, semiππΏπ βcontinuous functions etc. in fine-topological space and investigated their properties. Also defined ππΏπ βseperation axioms in fine-topological space. 2. Preliminaries Throughout this paper, (π, π) and (π, π) means topological spaces on which no separation axioms are assumed. For a subset A of a space X the closure and interior of A with respect to π are denoted by ππ(π΄) and πππ‘ π΄ . We use the following definitions: Definition 2.1 A subset A of a topological space X is said to be regular open if π΄ = πππ‘(ππ(π΄)) and the compliment of regular open set is called regular closed (cf. [17]). Definition 2.2 The largest regular open set contained in A is called the πΏ βinterior of A and is denoted by πΏ β πππ‘ π΄ (cf. [17]). Definition 2.3 The set A of X is called πΏ βopen if π΄ = πΏ β πππ‘ π΄ . The complement of πΏ βopen is called πΏ βclosed (cf. [17]). Definition 2.4 A subset A of a space (π, π) is called 1) Semi-open if π΄ β ππ πππ‘ π΄ (cf. [11]). 2) πΌ βopen if π΄ β πππ‘(ππ(πππ‘(π΄))) (cf. [8]). Definition 2.5 The largest semi-open set contained in A is called the semi-interior of A and is denoted by π πππ‘ π΄ (cf. [6]). Definition 2.6 The smallest semi-closed set containing the set A is called semi-closure of A and is denoted by π ππ(π΄)(cf. [6]). 539 ISSN 2320-5407 International Journal of Advanced Research (2013), Volume 1, Issue 10, 539-545 Definition 2.7 A subset A of X is called ππΏπ βclosed [1] if π ππ π΄ β π whenever π΄ β π and U is πΏ βopen in X. The family of all ππΏπ βclosed subsets of the space X is denoted by πΊπΏπ π π . Definition 2.8 The smallest ππΏπ βclosed set containing the set A is called ππΏπ βclosure of A and is denoted by ππΏπ β ππ π΄ . A set A is ππΏπ βclosed if ππΏπ β ππ π΄ = π΄ (cf. [1]). Definition 2.9 The largest ππΏπ βopen set contained in A is called ππΏπ βinterior of A and is denoted by ππΏπ β πππ‘ π΄ . A set A is ππΏπ βopen if ππΏπ β πππ‘ π΄ = π΄ (cf. [1]). Definition 2.10 A function π: π β π is called 1) Semi-continuous [11] if π β1 (π) is semi-closed in X for every closed set V in Y. 2) ππΏπ βcontinuous [2] if π β1 (π) is ππΏπ β-closed in X for every closed set V in Y. 3) Semi ππΏπ βcontinuous [2] if π β1 (π) is ππΏπ -closed in X for every semi-closed set V in Y. 4) ππΏπ βirresolute [7] if π β1 (π) is ππΏπ -closed in X for every ππΏπ βclosed set V in Y. 5) ππΏπ βopen [3] if π(π) is ππΏπ βopen in Y for every closed set V in X. 6) πππΏπ βopen [3] if π(π) is ππΏπ βopen in Y for every semi-open set V in X. 7) ππ’ππ π β ππΏπ βopen [4] if π(π) is open in Y for every ππΏπ βopen set V in X. 8) Strongly ππΏπ βopen [4] if π(π) is ππΏπ βopen in Y for every ππΏπ βopen set V in X. 9) Semi-closed [9] if π(π) is semi-closed in Y for every closed set V in X. 10) Pre-closed [10] if π(π) is closed in Y for every semi-closed set V in X. Definition 2.11 A topological space X is said to be ππΏπ β π0 space if for each pair of distinct points x and y of X, there exists a ππΏπ βopen set containing one point but not the other (cf. [5]). Definition 2.12 A topological space X is said to be ππΏπ β π1 space if for any pair of distinct point x and y, there exists a ππΏπ βopen sets G and H such that π₯ β πΊ, π¦ β πΊ and π₯ β π», π¦ β π» (cf. [5]). Definition 2.13 A topological space X is said to be ππΏπ β π2 space if for any pair of distinct points x and y, there exists disjoint ππΏπ βopen sets G and H such that π₯ β πΊ and π¦ β π» (cf. [5]). Definition 2.14 A topological space X is said to be ππΏπ βregular if for each closed set F and each point π₯ β πΉ, there exist disjoint ππΏπ βopen sets U and V such that π₯ β π and Fβ V (cf. [5]). Definition 2.15 Let (X, Ο) be a topological space we define Ο (π΄πΌ ) = ππΌ (say) = {πΊπΌ (β X) : πΊπΌ β© π΄πΌ = π, for π΄πΌ β π and π΄πΌ β π, π, for some Ξ± β J, where J is the index set.} Now, we define ππ = {π, π,βͺ πΌ βπ½ {ππΌ }} The above collection ππ of subsets of X is called the fine collection of subsets of X and (X, Ο, ππ ) is said to be the fine space X generated by the topology Ο on X (cf. [16]). Definition 2.16 A subset U of a fine space X is said to be a fine-open set of X, if U belongs to the collection ππ and the complement of every fine-open sets of X is called the fine-closed sets of X and we denote the collection by πΉπ (cf. [16]). Definition 2.17 Let A be a subset of a fine space X, we say that a point x β X is a fine limit point of A if every fineopen set of X containing x must contains at least one point of A other than x (cf. [16]). Definition 2.18 Let A be the subset of a fine space X, the fine interior of A is defined as the union of all fine-open sets contained in the set A i.e. the largest fine-open set contained in the set A and is denoted by ππΌππ‘ (cf. [16]). Definition 2.19 Let A be the subset of a fine space X, the fine closure of A is defined as the intersection of all fineclosed sets containing the set A i.e. the smallest fine-closed set containing the set A and is denoted by πππ (cf. [16]). Definition 2.20 A function f : (X, Ο, ππ ) β (Y, Ο β², ππβ² ) is called fine-irresolute (or f-irresolute) if π β1 (π ) is fineopen in X for every fine-open set V of Y (cf. [16]). 3. Fine ππΉπ βopen sets and Fine ππΉπ βcontinuous functions In this section, we have defined fine ππΏπ βopen sets and fine ππΏπ βcontinuous functions. Definition 3.1 The largest f-regular open set contained in A is called fine-πΏ βinterior of A and is denoted by π β πΏ β πΌππ‘ π΄ . Definition 3.2 The smallest f-regular closed set containing A is called fineβπΏ βclosure of A and is denoted by π β πΏ β ππ π΄ . Definition 3.3 A subset A of X is said to be fineβππΏπ βclosed if π β π β ππ π΄ β π whenever π΄ β π and π is fine β πΏ βopen in X. The family of all π β ππΏπ βclosed sets is denoted by πΉπΊπΏππΆ π . Definition 3.4 The intersection of all fineβππΏπ βclosed sets containing a set A is called π β ππΏπ βclosure of A and is denoted by π β ππΏπ β ππ(π΄). A set π΄ is π β ππΏπ βclosed iff π β ππΏπ β ππ π΄ = π΄. Definition 3.5 The union of all πππΏπ βopen sets contained in A is called π β ππΏπ βinterior of A and is denoted by π β ππΏπ β πΌππ‘ π΄ . A set A is π β ππΏπ βopen if and only if π β ππΏπ β πΌππ‘ π΄ = π΄. 540 ISSN 2320-5407 International Journal of Advanced Research (2013), Volume 1, Issue 10, 539-545 Definition 3.6 A function π: π β π is called fineβππΏπ βcontinuous if π β1 (π) is fineβππΏπ βclosed in X for every fine-closed set V in Y. Definition 3.7 A function π: π β π is called fine-semiβππΏπ βcontinuous if π β1 (π) is fineβππΏπ βclosed in X for every fine-semi-closed set V in Y. Definition 3.8 A function π: π β π is called fineβππΏπ βirresolute if π β1 (π) is fineβππΏπ βclosed in X for every fineβππΏπ βclosed set V in Y. Definition 3.9 A function π: π β π is called fineβππΏπ βopen if π(π) is fineβππΏπ βopen in Y for every fine-open set V in X. Definition 3.10 A function π: π β π is called fineβπππΏπ βopen if π(π) is fineβππΏπ βclosed in Y for every finesemi-open set V in X. Definition 3.11 A function π: π β π is called fine-quasi- ππΏπ βopen if π(π) is fine-open in Y for every πππΏπ βopen set V in X. Definition 3.12 A function π: π β π is called fine-stronglyβππΏπ βopen if π(π) is fineβππΏπ βclosed in Y for every fine-ππΏπ -open set V in X. Definition 3.13 A function π: π β π is called fine-semi-closed if π(π) is fine-semi-closed in Y for every fineclosed set V in X. Definition 3.14 A function π: π β π is called fine-pre closed if π(π) is fine-closed in Y for every fine-semi-open set V in X. Remark 3.15 1) Every ππΏπ βopen set is fine open and every ππΏπ βclosed set if fine-closed. 2) Every semi-closed and pre-closed set is fine closed. Remark 3.17 By the Definition 2.20, Definition 2.10 and Remark 3.15, we conclude the following: β Semi-continuous β ππΏπ βcontinuous β Semi-ππΏπ -continuous β ππΏπ -irresolute Fine-irresolute mapping β ππΏπ βopen β π ππΏπ βopen β ππ’ππ π β ππΏπ βopen β Strongly ππΏπ βopen β Semi-open β Pre-closed 4. Fine ππΉπ βseparaton axioms In this section, we introduce and study weak separation axioms such as π β ππΏπ β π 0 , π β ππΏπ β π 1 and π β ππΏπ β π 2 spaces and obtain some of their properties. Definition 4.1 A topological space X is said to be π β ππΏπ β π 0 space if for each pair of distinct points x and y of X, there exists a π β ππΏπ βopen set containing one point but not the other. Theorem 4.1 A topological space X is a π β ππΏπ β π 0 space if and only if π β ππΏπ βclosures of distinct points are distinct. Proof. Let x and y be distinct points of X. Since, X is π β ππΏπ β π 0 space, there exists a π β ππΏπ βopen set G such that π₯ β πΊ and π¦ β πΊ . Consequently, π β πΊ is a π β ππΏπ βclosed set containing y but not x. But, π β ππΏπ β ππ (π¦ )is the intersection of all π β ππΏπ βclosed set containing y. Hence, π¦ β π β ππΏπ β ππ (π¦ ) but π₯ β π β ππΏπ β ππ (π¦ ) as π₯ β π β πΊ . Therefore, π β ππΏπ β ππ π₯ β π β ππΏπ β ππ π¦ . Conversely, let π β ππΏπ β ππ π₯ β π β ππΏπ β ππ π¦ for π₯ β π¦ . Then, there exists at least one point π§ β π such that π§ β π β ππΏπ β ππ (π₯ ) but π§ β π β ππΏπ β ππ π¦ . We claim π₯ β π β ππΏπ β ππ (π¦ ), because if π₯ β π β ππΏπ β ππ (π¦ ) then {π₯ } β π β ππΏπ β ππ π¦ β π β ππΏπ β ππ π₯ β π β ππΏπ β ππ π¦ . So π§ β π β ππΏπ β ππ π¦ , which is a contradiction, hence π₯ β π β ππΏπ β ππ π¦ β π₯ β π β πππΏπ ππ π¦ , which is a π β ππΏπ βopen set containing x but not y. Hence, X is a π β ππΏπ β π 0 space. Theorem 4.2 If π : π β π is a bijection strongly π β ππΏπ βopen and X is π β ππΏπ β π 0 space, then Y is also π β ππΏπ β π 0 space. Proof. Let π¦ 1 and π¦ 2 be two distinct points of Y. Since f is bijection there exist distinct points π₯ 1 and π₯ 2 of X such that π π₯ 1 = π¦ 1 and π π₯ 2 = π¦ 2 . Since, X is π β ππΏπ β π 0 space there exists a π β ππΏπ βopen set G such that π₯ 1 β πΊ and π₯ 2 β πΊ . Therefore, π¦ 1 = π π₯ 1 β π (πΊ ) and π¦ 2 = π (π₯ 2 ) β π (πΊ ). Since, f being strongly fineππΏπ βopen function, f(G) is fine-ππΏπ βopen in Y. Thus, there exists a π β ππΏπ βopen set f(G) in Y such that π¦ 1 β π (πΊ ) and π¦ 2 β π πΊ . Therefore, Y is π β ππΏπ β π 0 space. 541 ISSN 2320-5407 International Journal of Advanced Research (2013), Volume 1, Issue 10, 539-545 Definition 4.2 A topological space X is said to be π β ππΏπ β π 1 space if for any pair of distinct points x and y there exist a π β ππΏπ βopen sets G and H such that π₯ β πΊ , π¦ β πΊ and π₯ β π» , π¦ β π» . Theorem 4.3 A topological space X is π β ππΏπ β π 1 space if and only if singletons are π β ππΏπ βclosed sets. Proof. Let X be a π β ππΏπ β π 1 space and π₯ β π . Let π¦ β π β {π₯ }. Then, for π₯ β π¦ , there exists π β ππΏπ βopen set π π¦ such that π¦ β π π¦ and π₯ β π π¦ . Consequently, π¦ β π π¦ β π β {π₯ }. That is π β {π₯ } =βͺ {π π¦ : π¦ β π β {π₯ }}, which is π β ππΏπ βopen. Hence, {π₯ } is π β ππΏπ βclosed set. Conversely, suppose {π₯ } is π β ππΏπ βclosed set for every π₯ β π . Let x and π¦ β π with π₯ β π¦ . Now, π₯ β π¦ β π¦ β π β {π₯ }. Hence, π β {π₯ } is π β ππΏπ βopen set containing y but not x. Similarly, π β {π¦ } is π β ππΏπ βopen set containing x but not y. Therefore, π is π β ππΏπ β π 1 space. Theorem 4.4 The property being π β ππΏπ β π 1 space is preserved under bijection and strongly fine-ππΏπ βopen function. Proof. Let π : π β π be bijection and strongly fineβππΏπ βopen function. Let X be a π β ππΏπ β π 1 space and π¦ 1 and π¦ 2 be two distinct points of Y. Since f is bijective there exists distinct points π₯ 1 , π₯ 2 of X such that π¦ 1 = π (π₯ 1 ) and π¦ 2 = π (π₯ 2 ). Now, X being a π β ππΏπ β π 1 space, there exist π β ππΏπ βopen sets G and H such that π₯ 1 β πΊ , π₯ 2 β πΊ and π₯ 1 β π» , π₯ 2 β π» . Therefore, π¦ 1 = π π₯ 1 β π (πΊ ) but π¦ 2 = π π₯ 2 β π (πΊ ) and π¦ 2 = π π₯ 2 β π (π» ) and π¦ 1 = π π₯ 1 β π π» . Now, f being strongly π β ππΏπ βopen, f(G) and f(H) are π β ππΏπ βopen subsets of Y such that π¦ 1 β π (πΊ ) but π¦ 2 β π (πΊ ) and π¦ 2 β π (π» ) and π¦ 1 β π π» . Hence, Y is π β ππΏπ β π 1 space. Theorem 4.5 If π : π β π be π β ππΏπ βcontinuous injection and Y be π 1 , then X is π β ππΏπ β π 1 . Proof. If π : π β π be π β ππΏπ βcontinuous injection and Y be π 1 . For any two distinct points π₯ 1 , π₯ 2 of X there exist distinct points π¦ 1 , π¦ 2 of Y such that π¦ 1 = π (π₯ 1 ) and π¦ 2 = π π₯ 2 . Since, Y is π 1 βspace there exist fineopen sets U and V in Y such that π¦ 1 β π , π¦ 2 β π and π¦ 1 β π , π¦ 2 β π . That is π₯ 1 β π β1 π , π₯ 1 β π β1 (π ) and π₯ 2 β π β1 π , π₯ 2 β π β1 π . Since, f is π β ππΏπ βcontinuous, π β1 π , π β1 (π ) are π β ππΏπ βopen sets in X. Thus, for two distinct points π₯ 1 , π₯ 2 of X there exist π β ππΏπ βopen sets π β1 π and π β1 π such that π₯ 1 β π β1 π , π₯ 1 β π β1 (π ) and π₯ 2 β π β1 π , π₯ 2 β π β1 π . Therefore, X is π β ππΏπ β π 1 space. Theorem 4.6 If π : π β π be π β ππΏπ βirresolute injective and Y π β ππΏπ β π 1 space, then X is π β ππΏπ β π 1 space. Proof. Let π₯ 1 , π₯ 2 be pair of distinct points in X. Since f is injective there exist distinct points π¦ 1 , π¦ 2 of Y such that π¦ 1 = π (π₯ 1 ) and π¦ 2 = π π₯ 2 . Since, Y is π β ππΏπ β π 1 space there exist π β ππΏπ βopen sets U and V in Y such that π¦ 1 β π , π¦ 2 β π and π¦ 1 β π , π¦ 2 β π . That is π₯ 1 β π β1 π , π₯ 1 β π β1 (π ) and π₯ 2 β π β1 π , π₯ 2 β π β1 π . Since f is π β ππΏπ βirresolute π β1 π , π β1 (π ) are π β ππΏπ βopen sets in X. Thus for two distinct points π₯ 1 , π₯ 2 of X there exist π β ππΏπ βopen sets π β1 π and π β1 (π ) such that π₯ 1 β π β1 π , π₯ 1 β π β1 (π ) and π₯ 2 β π β1 π , π₯ 2 β π β1 π . Therefore, X is π β ππΏπ β π 1 space. Definition 4.3 A topological space X is said to be π β ππΏπ β π 2 space if for any pair of distinct points x and y, there exist disjoint π β ππΏπ βopen sets G and H such that π₯ β πΊ and π¦ β π» . Theorem 4.7 If π : π β π is π β ππΏπ βcontinuous injection and Y is π 2 then X is π β ππΏπ β π 2 space. Proof. Let π : π β π be π β ππΏπ βcontinuous injection and Y is π 2 . For any two distinct points π₯ 1 , π₯ 2 of X there exist distinct points π¦ 1 , π¦ 2 of Y such that π¦ 1 = π (π₯ 1 ) and π¦ 2 = π π₯ 2 . Since, Y is π 2 space there exist disjoint fine-open sets U and V in Y such that π¦ 1 β π and π¦ 2 β π . That is π₯ 1 β π β1 (π ) and π₯ 2 β π β1 π . Since, f is fineππΏπ βcontinuous π β1 (π ) and π β1 (π ) are π β ππΏπ βopen sets in X. Further f is injective, π β1 π β© π β1 π = π β1 π β© π = π β1 π = π . Thus, for two disjoint points π₯ 1 , π₯ 2 of X there exist disjoint π β ππΏπ βopen sets π β1 (π ) and π β1 (π ) such that π₯ 1 β π β1 (π ) and π₯ 2 β π β1 (π ). Therefore, X is π β ππΏπ β π 2 space. Theorem 4.8 If π : π β π is π β ππΏπ βirresolute injective function and Y is π β ππΏπ β π 2 space then X is π β ππΏπ β π 2 space. Proof. Let π₯ 1 , π₯ 2 be pair of distinct points in X. Since f is injective there exist distinct points π¦ 1 , π¦ 2 of Y such that π¦ 1 = π π₯ 1 , π¦ 2 = π (π₯ 2 ). Since Y is π β ππΏπ β π 2 space there exist disjoint π β ππΏπ βopen sets U and V in Y such that π¦ 1 β π and π¦ 2 β π in Y. That is π₯ 1 β π β1 (π ) and π₯ 2 β π β1 π . Since, f is π β ππΏπ βirresolute injective π β1 π , π β1 (π ) are distinct π β ππΏπ βopen sets in X. Thus, for two distinct points π₯ 1 , π₯ 2 of X there exist disjoint π β ππΏπ βopen sets π β1 (π ) and π β1 (π ) such that π₯ 1 β π β1 (π ) and π₯ 2 β π β1 π . Therefore, X is π β ππΏπ β π 2 space. Theorem 4.9 In any fine-topological space the following are equivalent: 1) X is π β ππΏπ β π 2 space. 2) For each π₯ β π¦ , there exists a π β ππΏπ βopen set U such that π₯ β π and π¦ β π β ππΏπ β ππ π . 542 ISSN 2320-5407 International Journal of Advanced Research (2013), Volume 1, Issue 10, 539-545 3) For each π₯ β π , {π₯ } =β© { π β ππΏπ β ππ π : π is π β ππΏπ βopen set in X and π₯ β π . Proof. 1)βΉ2) Assume (1) holds. Let π₯ β π and π₯ β π¦ , then there exist disjoint π β ππΏπ βopen sets U and V such that π₯ β π and π¦ β π . Clearly, π β π is π β ππΏπ βclosed set. Since, π β© π = π , π β π β π . Therefore, π β ππΏπ β ππ π β π β ππΏπ β ππ π β π = π β π . Now, π¦ β π β π β π¦ β π β ππΏπ β ππ π . 2)βΉ3) For each π₯ β π¦ , there exust a π β ππΏπ βopen set U such that π₯ β π and π¦ β π β ππΏπ β ππ π . So, π¦ β β© {π β ππΏπ β ππ π : π is πππΏπ βopen in X and π₯ β π } = {π₯ }. 3)βΉ1) Let π₯ , π¦ β π and π₯ β π¦ . By hypothesis there exist a πππΏπ βopen set U such that π₯ β π and π¦ β πππΏπ β ππ π . This implies there exists a πππΏπ βclosed set V such that π¦ β π . Therefore, π¦ β π β π and π β π is πππΏπ βopen set. Thus, there exist two disjoint πππΏπ βopen sets U and X-V such that π₯ β π and π¦ β π β π . Therefore X is πππΏπ β π 2 space. 5. FineβππΉπ βregular space In this section, we introduce and study πππΏπ βregular space and some of their properties. Definition 5.1 A topological space X is said to be πππΏπ βregular space if for each fine-cloed set F and each point π₯ β πΉ , there exist disjoint πππΏπ βopen sets U and V such that π₯ β π and πΉ β π . Theorem 5.1 Every πππΏπ βregular π0 space isπππΏπ β π 2. Proof. Let π₯ , π¦ β π such that π₯ β π¦ . Let X be a π 0 space and V be a fine βopen set which contains x but not y. Then, X-V is a closed set containing y but not x. Now, by πππΏπ βregularity of X there exist disjoint πππΏπ βopen sets U and W such that π₯ β π and π β π β π. Since, π¦ β π β π , π¦ β π. Thus, for π₯ , π¦ β π with π₯ β π¦ , there exist disjoint open sets U and W such that π₯ β π andπ¦ β π. Hence, X is πππΏπ β π 2. Theorem 5.2 If π : π β π is fine-continuous bijective πππΏπ βopen function and X is a fine-regular space, then Y is πππΏπ βregular. Proof. Let F be a fine-closed set in Y and π¦ β πΉ . take π¦ = π (π₯ ) for some π₯ β π . Since f is continuous surjective π β1 (πΉ ) is fine-closed set in X and π₯ β π β1 πΉ . Now, since X is fineβregular, there exist disjoint fine-open sets U and V such that π₯ β π and π β1 πΉ β π . That is π¦ = π π₯ β π (π ) and πΉ β π π . Since, f is πππΏπ βopen function f(U) and f(V) are πππΏπ βopen sets in Y and f is bijective. π π β© π π = π π β© π = π π = π . Therefore, Y is πππΏπ βregular. Theorem 5.3 If π : π β π is πππΏπ βcontinuous, fine-closed injection and Y is fine-regular, then X is πππΏπ βregular. Proof. Let F be a fine-closed set in X and π₯ β πΉ . Since, f is closed injection f(F) is fine-closed set in Y such that π π₯ β π πΉ . Now, Y is fine-regular there exist disjoint fine-open sets G and H such π π₯ β πΊ and π πΉ β π» . This implies π₯ β π β1 (πΊ ) and πΉ β π β1 π» . Since, f is πππΏπ βcontinuous, π β1 (πΊ ) and π β1 (π» ) are β1 β1 πππΏπ βopen sets in X. Further π πΊ β© π π» = π . Hence, X is πππΏπ βregular. Theorem 5.4 If π : π β π is fine-semi-ππΏπ βcontinuous, fine-closed injection and Y is fine-semi regular, then X is πππΏπ βregular. Proof. Let F be a fine-closed set in X and π₯ β πΉ . Since, f is fine-closed injection f(F) is fine-closed set in Y such that π π₯ β π πΉ . Now, Y is fine-semi-regular, there exist disjoint fine-semi-open sets G and H such that π π₯ β πΊ and π πΉ β π» . This implies π₯ β π β1 (πΊ ) and πΉ β π β1 π» . Since, f is fine-semi-ππΏπ βcontinuous π β1 (πΊ ) and π β1 (π» ) are πππΏπ βopen sets in X. Further, π β1 πΊ β© π β1 π» = π . Hence, X is πππΏπ βregular. Theorem 5.5 If π : π β π is πππΏπ βirresolute, fine-closed injection and Y is πππΏπ βregular then X is πππΏπ βregular. Proof. Let F be a fine-closed set in X and π₯ β πΉ . Since, f is fine-closed injection f(F) is fine-closed set in Y such that π (π₯ ) β π (πΉ ). Now, Y is πππΏπ βregular, there exist diajoint πππΏπ βopen sets G and H such that π π₯ β πΊ and π πΉ β π» . This implies π₯ β π β1 (πΊ ) and πΉ β π β1 π» . Since, f is πππΏπ βirresolute π β1 (πΊ ) and π β1 (π» ) β1 β1 are πππΏπ βopen sets in X. Further π πΊ β© π π» = π . Hence, X is πππΏπ βregular. 6. Fine-ππΉπ βnormal space In this section, we introduce and study πππΏπ βnormal spaces and some of their properties. Definition 6.1 A topological space X is said to be πππΏπ βnormal if every pair of disjoint fine-closed sets E and F of X there exist disjoint πππΏπ βopen sets U and V such that πΈ β π and πΉ β π . Theorem 6.1 The following statements are equivalent for a fine-topological space X: 1) X is πππΏπ βnormal. 2) For each fine-closed set A and for each fine-open set U containing A, there exist a πππΏπ βopen set V containing A such that πππΏπ β ππ π β π . 543 ISSN 2320-5407 International Journal of Advanced Research (2013), Volume 1, Issue 10, 539-545 3) For each pair of disjoint fine-closed sets A and B there exists a πππΏπ πππΏπ β π π π β© π΅ = π. βopen set U containing A such that Proof. 1)βΉ2) Let A be a fine-closed set and U be a fine-open set containing A. Then, π΄ β© π β π = π and therefore they are disjoint fine-closed sets in X. Since, X is πππΏπ βnormal, there exist disjoint πππΏπ βopen sets V and W such that π΄ β π , π β π β π that is π β π β π . Now π β© π = π , implies π β π β π. Therefore, πππΏπ β ππ π β πππΏπ β ππ π β π = π β π, because X-W is πππΏπ βclosed set. Thus, π΄ β π β πππΏπ ππ π β π β π β π . That is π΄ β π β πππΏπ ππ π β π. 2)βΉ3) Let A and B be disjoint fine-closed sets in X, then π΄ β π β π΅ and X-B is a fine-open set containing A. By (2), there exists a πππΏπ βopen set U such that π΄ β π and πππΏπ β ππ π β π β π΅ , which implies πππΏπ β ππ π β© π΅ = π . 3)βΉ 1) Let A and B be disjoint fine-closed sets in X. By (3) there exist a πππΏπ βopen set U such that π΄ β π and πππΏπ β ππ π β© π΅ = π or π΅ β π β πππΏπ β ππ (π . ) Now, U and π β πππΏπ β ππ (π ) are disjoint πππΏπ βopen sets of X such that π΄ β π and π΅ β π β π ππΏπ β ππ π . Hence, X is πππΏπ βnormal. Theorem 6.2 If X is fine semi-normal, then the following statements are true: 1) For each π β πΏ βclosed set A and every π β ππΏπ βopen set B such that π΄ β π΅ there exists a fine-semiopen set U such that π΄ β π β π β π β ππ π β π΅ . 2) For every πππΏπ βclosed set A and every π β πΏ βopen set containing A, there exists a fine-semi-open set containing U such that π΄ β π β π π β ππ π β π΅ . Proof. 1) Let A be a π β πΏ βclosed set and B be a πππΏπ βopen set such that π΄ β π΅ . Then, π΄ β© π β π΅ = π . Since, A is a π β π βclosed set and π β π΅ be a πππΏπ βclosed, by Theorem 6.1, there exists fine-semiopen sets U and V such that π΄ β π , π β π΅ β π and π β© π = π . Thus, π΄ β π β π β π β π΅ . Since, π β π is fine-semi-closed ππ ππ π β π β π . Therefore, π΄ β π β π π β ππ π β π΅ . 2) Let A be a πππΏπ βclosed set and B be a π β πΏ βopen set such that π΄ β π΅ . Then, π β π΅ β π β π΄ . Since, X is fine-semi-normal and X-A is a πππΏπ βopen set containing π β πΏ βclosed set π β π΅ , by(1) there exists a fine-semi-open set G such that π β π΅ β πΊ β π π ππ πΊ β π β π΄ . That is, π΄ β π β π π β ππ πΊ β π β πΊ β π΅ . Let π = π β π π β ππ πΊ , then U is fine-semi-open set and π΄ β π β π π β ππ π β π΅ . 7. Conclusion The author has defined strongly ππΏπ βcontinuous functions, pre-ππΏπ βcontinuous functions, semiβcontinuous functions etc. in fine-topological space and investigated their properties also defined βseperation axioms in fine-topological space. This concept may be useful in Quantum physics, Quantum Mechanics, Quantum gravity etc. ππΏπ ππΏπ 8. Acknowledgement I would like to thank to my βGURUβ Prof. P. V. Jain madam (Professor, Department of Mathematics and Computer Science, Rani Durgawati University, Jabalpur, M.P., India) for her valuable suggestion and help. References [1] Benchali S. S. and Umadevi I. Neeli, (2011): Generalized πΏ βsemi-closed sets in topological spaces, Int. J. Appl. Math., 24, 21-38. [2] Benchali S. S. and Umadevi I. Neeli, πΏππ βcontinuous functions in topological spaces, communicated. [3] Benchali S. S. and Umadevi I. Neeli, πΏππ Interdidciplinary stud. Accapted. [4] Benchali S. S. and Umadevi I. 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