Ratio Mathematica Volume 45, 2023 Micro Sp-Open Sets in Micro Topological Spaces M. Maheswari1 S. Dhanalakshmi2 N. Durgadevi3 Abstract In this paper, a new class of open sets called Micro Sp- Open sets in Micro topological spaces are introduced and its fundamental properties are analyzed. Also, some operations on Micro Sp-open sets are investigated. Keywords: Micro-open, Micro-Semi open, Micro-Pre-open, Micro-Pre closed and Micro Sp-open, Micro Sp-closed. 2010 Mathematics Classification:46S40, 34A07, 03E724 1 Research Scholar, Department of Mathematics, Sri Parasakthi College for Women, Courtallam. (Affiliated to Manonmaniam Sundaranar University, Tirunelveli, Tamilnadu). 2 Research Scholar, Department of Mathematics, Sri Parasakthi College for Women, Courtallam. (Affiliated to Manonmaniam Sundaranar University, Tirunelveli, Tamilnadu). 3 Assistant Professor, Department of Mathematics, Sri Parasakthi College for Women, Courtallam (Affiliated to Manonmaniam Sundaranar University, Tirunelveli, Tamilnadu). Email: durgadevin@sriparasakthicollege.edu.in 4Received on July 20, 2022. Accepted on October 15, 2022. Published on January 30, 2023. doi: 10.23755/rm. v45i0.985. ISSN: 1592-7415. eISSN: 2282-8214. ©The Authors. This paper is published under the CC-BY license agreement. 90 mailto:durgadevin@sriparasakthicollege.edu.in M. Maheswari, S. Dhanalakshmi and N. Durgadevi 1. Introduction The concept of Nano topology was introduced by Lellis Thivagar [3] which is in terms of the lower and upper approximations and the boundary region of a subset of an universe. The notion of approximations and boundary region of a set was originally proposed by Pawlak [4] in order to introduce the concept of rough set theory. Chandrasekar [5] introduced the concept of micro topology which is a simple extension of Nano topology and he also studied the concepts of Micro pre-open and Micro semi- open sets. In 2010, Shareef introduced the class of semi open sets called Sp-open sets. Chandrasekar and Swathi [1] introduced Micro -open in micro topological space. In this paper a new class of sets in Micro topological spaces called Micro Sp-Open set is introduced and some of its properties are derived. 2. Preliminaries Definition 2.1. [4] Let U be a non-empty finite set of objects called the universe and R be an equivalence relation on U named as the indiscernibility relation. Elements belonging to the same equivalence class are said to be indiscernible with one another. The pair (U, R) is said to be the approximation space, Let X  U, Then (i) The Lower approximation of X with respect to R is the set of all objects which can be certain classified as X with respect to R and is denoted by LR(X). That is LR(X) =  UX XR  )({ : R(X)  X} where R(X) denotes the equivalence class determined by X U. (ii) The upper approximation of X with respect to R is the set of all objects, which can be possibly classified as X with respect to R and it is denoted by UR(X). That is UR(X) = U UX XR  )({ : R(X) X}. (iii) The Boundary region of X with respect to R is the set of all objects, which can be classified as neither as X nor as not-X with respect to R and is denoted by BR(X). That is BR(X) = UR(X) – LR(X). Definition2.2. [3] Let R be an equivalence relation on the universe U and R(X) = {U,, LR(X), UR(X), BR(X)} and xU. Then R(X)satisfies the following axioms. (i) U and R(X). (ii) The union of the elements of any sub collection of R(X) is in R(X). (iii) The intersection of the elements of any finite subcollection of R(X) is in R(X). That is,R(X) is a topology on U called the Nano topology on U with respect to X. Thus (U, R(X)) as called as Nano topological space. The elements of R(X) are called as Nano open sets. A subset F of U is nano closed if its complement is nano open. Definition 2.3. [5] Let (U,R(X)) be a Nano topological space. Then R(X) = {N ∪ (N):N, NR(X)} and R(X)) is called the Micro topology in U with respect to X. The triplet (U,R(X), R(X)) is called Micro topological space and then elements of 91 Micro Sp-Open Sets in Micro Topological Spaces R(X) are called Micro open sets and the complement of a Micro open set is called a Micro closed set. Definition 2.4. [1] Let (U,R(X), R(X))be a micro topological space and AU. Then A is called (i) Micro -open if A  Mic-Int (Mic-Cl (Mic-Int (A))) (ii) Micro pre-open if A  Mic-Int (Mic-Cl (A)). (iii) Micro semi-open if A  Mic-Cl (Mic-Int (A)). Definition 2.5. [5] Let (U, R(X), R(X)) be a Micro topological space. Let A and B be any two subsets of U. Then (i) A is Micro open set if and only if Mic-Int(A)=A. (ii) A is micro closed set if an only if Mic-Cl(A) = A. (iii) Mic-Int(U\A) = U\Mic-Cl(A). (iv) Mic-Cl(U\A) =U\Mic-Int(A). 3. Micro Sp-Open Sets Definition 3.1. Let (U, R(X), R(X)) be a Micro topological space and A  U. Then A is said to be Micro SP-open (briefly Mic SP-open) if for each xAMic-SO (U, X), there exists a Micro pre-closed set F such that XF  A. The set of all Micro SP-open sets is denoted by Mic SP-O (U, X). Definition 3.2. Let (U, R(X), R(X)) be a Micro topological space. A subset B of U is called Micro SP-closed (briefly Mic SP-closed) if and only if its complement is Micro SP-open and Mic SP-CL (U, X) denotes the collection of all Micro Sp-closed sets. Example 3.3. Let U = {a, b, c, d} with U|R = {{a}, {c}, {b, d}}, X = {b, d}  U then R(X) = {U, , {b, d}}. If  = {a}. Then R(X) = {U,, {a}, {b, d}, {a, b, d}} and Micro Sp-open sets are {U, , {a, c}, {b, d}, {b, c, d}} Remark 3.4. Every Micro Sp-open set is a Micro semi open set but the converse need not always be true as shown from the following example. Example 3.5. In example 3.3, Mic-SO(U, X) = {U, , {a},{a, c}, {b, d}, {a, b, d}, {b, c, d}}.Mic SP-O (U, X) = {U,, {a, c}, {b, d}, {b, c, d}} the set {a, b, d} is Micro semi- open but not Micro SP-open. Theorem 3.6. An arbitrary union of any family of Micro SP-open sets is Micro SP-open. Proof: Let {Ai: i} be a family of Mic SP-open sets. If xA, then for each x i i A  Mic-SO (U, X) there exists a Micro pre-closed set F such that x  F  Ai U i i A which 92 M. Maheswari, S. Dhanalakshmi and N. Durgadevi implies x  F  U i i A . Therefore U i i A is Micro SP-open. Remark 3.7. From the above Theorem 3.6, arbitrary intersection of Micro SP-closed sets of a Micro topological space is Micro SP-closed as shown by the following example. Example 3.8. In example 3.3, Mic SP-O (U, X) = {U,, {a, c}, {b, d}, {b, c, d}} and Mic Sp-CL (U, X) = {U, , {b, d}, {a, c}, {a}}. Here {b, d}  {b, c, d} = {b, d} which is a Micro SP-closed set. Remark 3.9.The intersection of any two Mic SP-open sets need not be a Mic SP-open set. From example 3.3, {a, c}, {b, c, d} are Mic SP-open sets but {a, c}  {b, c, d} = {c} is not a Mic SP-open set. Proposition 3.10. If a subset A of a Micro topological space (U, R(X), R(X)) is Mic SP- open. Then A is a Micro semi-open set and A is a union of Micro pre-closed sets. Proof: Let A Mic SP-O (U, X) and A  U. Then for each x  A  Mic-SO (U, X), there exists a Micro pre-closed set F containing x such that x  F  A. Thus A is a Micro semi-open set and also F  A which implies A is the union of Micro pre-closed sets. Remarks 3.11. The converse of the above Proposition 3.10 need not be true as shown in the following example. Example 3.12. In example 3.3, Mic-SO (U, X) = {U, , {a}, {a, c}, {b, d}, {a, b, d}, {b, c, d}} and Mic-PCL (U, X) = {U, , {b}, {c}, {d}, {a, c}, {b, c}, {c, d}, {a, b, c}, {a, c, d}, {b, c, d}}. Then {a, b, d}  Mic-SO (U, X) and {a, b, d} is not in the union of Micro Pre-closed sets, also {a, b, d}  Mic SP-O (U, X). 4. Operations on Micro SP-open Sets Definition 4.1. A point xU is said to be a Micro SP-interior point of A if there exists a Micro Sp-open set V containing x such that VA. The set of all Mic Sp-interior points of A is said to be Micro Sp-interior of A and is denoted by Mic Sp-Int(A). Definition 4.2. Let A be any subset of a Micro topological space (U, R(X), R(X)). Then a point xU is in the Micro Sp-closure of A if and only if A  H  for every H Mic Sp-O (U, X) containing x. The intersection of all Micro Sp-closed sets containing H is called the Micro Sp-closure of F and is denoted by Mic Sp-Cl (A). 93 Micro Sp-Open Sets in Micro Topological Spaces Theorem 4.3. Let A be any subset of a Micro topological space (U, R(X), R(X)). If a point x Mic Sp-Int (A), then there exists F Mic-PCl (U, X) containing x such that F  A. Proof: Suppose that x  Mic Sp-Int (A). Then V Mic Sp-O (U, X) containing x such V  A. since V Mic Sp-O (U, X), then there exists F Mic-PCl (U, X) containing x such that F  U  A. Hence x  F  A. Theorem 4.4. Let A be a subset of a Micro topological space (U,R(X), R(X)). If A  F   for every FMic-PCl (U, X) containing x, then x Mic Sp-Cl (A). Proof: Let V Mic Sp-O (U, X) containing x, then there exists a Micro pre-closed set F containing x such that FV. Since A  F , x  Mic Sp-Cl (A). Theorem 4.5. For any two subsets A and B of a Micro topological space (U, R(X), R(X)), the following properties are true. (i) Mic Sp-Int (Mic Sp-Int (A)) = Mic Sp-Int (A). (ii) Mic Sp-Int (A) = U – Mic Sp-Cl (U – A). (iii) If A  B, then Mic Sp-Int(A)  Mic Sp-Int (B) (iv) Mic Sp-Int (A) Mic Sp-Int (B)  Mic Sp-Int (A  B). (v) Mic Sp-Int (A  B)  Mic Sp-Int (A)  Mic Sp-Int (B). Proof: Obvious. The converse of (iii), (iv) and (v) of Theorem 4.5 need not be true as shown in the following example. Example 4.6.Consider U = {p, q, r, s, t} with U|R = {{p, q, r}, {s}, {t}}, X = {p, q}  U, then R(X) = {U,, {p, q, r}}. If  = {t}, then R(X) = {U,, {t}, {p, q, r}, {p, q, r, t}} and Mic Sp-O(U, X) = {U, , {s, t}, {p, q, r}, {p, q, r, s}}. (iii) If A = {s} and B = {q, r, t}, then Mic SP-Int ({s}) =  = Mic Sp-Int ({q, r, t}) but A  B. (iv) Let A = {p, q} and B = {r, s}, then Mic Sp-Int ({p, q})  Mic Sp-Int ({r, s}) =  =. But Mic Sp-Int ({p, q}  {r, s}) = Mic Sp-Int ({p, q, r, s}) = {p, q, r, s} which implies that Mic Sp-Int (A  B)  Mic Sp-Int (A)  Mic Sp-Int (B). (v) Consider A = {p, q, r, s} and B = {p, s, t}, then Mic Sp-Int ({p, q, r, s})  Mic Sp-Int ({p, s, t}) = {p, q, r, s}  {s, t} = {s}. But Mic Sp-Int ({p, s, t}  {p, q, r, s}) = Mic Sp-Int ({p, s}) =. Therefore, Mic Sp-Int (A)  Mic Sp-Int(B)  Mic Sp- Int(AB). Theorem 4.7. For any two subsets A and B of a Micro topological space (U, R(X), R(X)). the following properties are true. (i) Mic Sp-Cl (Mic Sp-Cl (A)) = Mic SP-Cl (A). (ii) Mic Sp-Cl (A) = U – Mic Sp-Int (U – A). (iii) If A  B, then Mic Sp-Cl (A)  Mic Sp-Cl (B) 94 M. Maheswari, S. Dhanalakshmi and N. Durgadevi (iv) Mic Sp-Cl(A)  Mic Sp-Cl(B)  Mic Sp-Cl(AB). (v) Mic Sp-Cl(AB)  Mic Sp-Cl(A)  Mic Sp-Cl(B). Proof: Obvious, The converse of (iii), (iv) and (v) of Theorem 4.7 need not be true as shown in the following example. Example 4.8. Consider U= {p, q, r, s, t} with U|R = {{p, q, r}, {s}, {t}}, X = {p, q}  U, then R(X) = {U,, {p, q, r}}. If  = {t}, then R(X) = {U, , {t}, {p, q, r}, {p, q, r, t}} and Mic Sp-O(U, X) = {U,, {s, t}, {p, q, r}, {p, q, r, s}}, Mic Sp-Cl (U, X) = {U, , {p, q, r}, {s, t}, {t}}. (iii) Let A = {p, t} and B = {q, s, t}, then Mic SP-Cl ({p, t}) = U and Mic Sp-Cl ({q, s, t}) = U but A  B. (iv) Let A = {p, q, r} and B = {t}, then Mic Sp-Cl ({p, q, r})  Mic Sp-Cl ({t}) = {p, q, r}  {t} = {p, q, r, t}. But Mic Sp-Cl ({p, q, r}  {t}) = Mic Sp-Cl ({p, q, r, t}) = U. Therefore Mic Sp-Cl (AB)  Mic Sp-Cl (A)  Mic Sp-Cl (B). (v)In general, for any closure operator Cl (F)  Cl (E) = Cl (F  E) and for most of The closure operators Cl (F)  Cl (E)  Cl (FE). In the case of Mic SP-closure operator, the equality sign need not hold for both the cases and it is justified by the following example. This obviously leads to the conclusion that Mic SP-closure operator is not a Kuratowski’s operator. For, let A = {r} and B = {p, t} then Mic SP-Cl ({r}) Mic SP-Cl ({p, t}) = {p, q, r}  U = {p, q, r}. But Mic Sp-Cl ({r} {p, t}) = Mic Sp- Cl()= . Hence Mic SP-Cl(A)  Mic SP-Cl(B)  Mic SP-Cl(AB). 5.Conclusion In this paper Mic SP -open sets and Mic SP -closed sets are defined and some of their properties are discussed. This shall be extended in the future Research with some applications. Acknowledgement It is our pleasant duty to thank referees for their useful suggestions which helped us to improve our manuscript. References [1] Chandrasekar. S and Swathi. G: Micro -open sets in Micro topological spaces, International Journal of Research in advent Technology, Vol.6, No. 10, October 2018. [2] Hariwan. Z. Ibrahim: Micro -open sets in Micro topology, Gen. Lett., 8(1) (2020),8-15. 95 Micro Sp-Open Sets in Micro Topological Spaces [3] Lellis Thivagar. M and Richard. C: On Nano Forms of Weakly Open Sets, International Journal of Mathematics and Statistics Invention, 1/1 (2013), 31-37. [4] Pawlak. Z: Rough Sets., International Journal of Information and Computer Sciences, 11(1982), 341-356. [5] Sakkraiveeranan Chandrasekar: On Micro Topological Spaces, Journal of New Theory 26(2019), 23-31. [6] Saravanakumar. D, Sathiyanandham. T and Shalini. V. C: NSp-open sets and NSp- closed sets in Nano Topological Spaces, International Journal of Pure and Applied Mathematics, Volume 113, No.12, 2017, 98-106. [7] Shareef. A.H: Sp-open sets, Sp-continuity and Sp-continuity and Sp-compactness in topological spaces, M.Sc. Thesis, College of Science, Sulaimani Univ., 2007. [8] Quays Hatem Imran: On Nano Semi Alpha Open Sets, Journal of Science and Arts, Year 17, No.2(39), pp 235-244, 2017. 96