EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 17, No. 4, 2024, 3156-3166 ISSN 1307-5543 – ejpam.com Published by New York Business Global On Micro Pre Operators in Micro Topological Spaces P. Sathishmohan1, S. Stanley Roshan1,∗, K. Rajalakshmi2, S. Brindha3, G. Poongothai1 1 Department of Mathematics, Kongunadu Arts and Science College(Autonomous), Coimbatore-29, Tamil Nadu, India 2 Department of Science and Humanities, Sri Krishna College of Engineering and Technology, Coimbatore-08, Tamil Nadu, India 3 Department of Mathematics, Government Arts and Science College, Mettupalayam-04, Tamil Nadu, India Abstract. The basic objective of this research work is to introduce and investigate the properties of micro pre-frontier, micro pre-exterior, micro pre-border, micro pre-kernel using the concept of frontier, exterior, border and kernel. 2020 Mathematics Subject Classifications: 54A05, 54A10 Key Words and Phrases: Micro pre-frontier, micro pre-exterior, micro pre-border, micro pre- kernel 1. Introduction Levine’s introduction of generalized closed sets in 1970 [3], providing a foundational framework for subsequent developments. Lellis Thivagar [1], further expanded this frame- work with the introduction of nano topology, utilizing approximations and boundary re- gions of a subset of a universe using an equivalence relation on it to define nano closed sets, nano-interior and nano-closure. The exploration of weak forms of nano open sets, such as nano α-open sets, nano semi-open sets, nano pre-open sets, and nano-β-open sets, was undertaken by many authors adding layers of complexity to the existing theories. In 2013, Antony Rex Rodgio et.al.,[8] defined the properties of β∗ open sets like frontier, exterior and border. In 2018, Sathishmohan et.al., [6] introduced some properties of nano pre-neighbourhoods in nano topology. In 2019, Chandrasekar [4], introduced the concept of micro topology which is a simple extension of nano topology, with a focus on micro pre- open and micro semi-open sets. Chandrasekar and Swathi [5], introduced micro α-open ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v17i4.5434 Email addresses: sathishmohan@kongunaducollege.ac.in (P. Sathishmohan), stanleyroshan20@gmail.com (S. Stanley Roshan), rajalakshmikandhasamy@gmail.com (K. Rajalakshmi), brindha.sagashra@gmail.com (S. Brindha), gpkpoongothai@gmail.com (G. Poongothai) https://www.ejpam.com 3156 Copyright: © 2024 The Author(s). (CC BY-NC 4.0) S. Stanley Roshan et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3156-3166 3157 sets and studied the basic properties. Recently in 2020, Hariwan Z.Ibrahim [9], introduced micro β-open sets in micro topological spaces. The aim of this paper is to introduce and investigate the properties of micro pre-frontier, micro pre-exterior, micro pre-border and micro pre-kernel using the notion of frontier, exterior, border and kernel to obtain their basic results. The basic definitions used in this paper is given below. Definition 1. [1] Let U be the universe, R be an equivalence relation on U and τR(X) = {U, ∅, LR(X), UR(X), BR(X)} where X ⊆ U . τR(X) satisfies the following axioms: (i) U ∈ τR(X) and ∅ ∈ τR(X). (ii) The union of elements of any sub collection of τR(X) is in τR(X). (iii) The intersection of the elements of any finite sub collection of τR(X) is in τR(X). That is, τR(X) forms a topology on U is called the nano topology on U with respect to X. {U, τR(X)} is called the nano topological space. Definition 2. [4] Let {U,τR(X)} is a nano topological space here µR(X) = {N ∪(N ′∩µ) : N,N ′ ∈ τR(X)} and called it micro topology of τR(X) by µ where µ /∈ τR(X). Definition 3. [4] The micro topology µR(X) satisfies the following axioms. (i) U ∈ µR(X) and ∅ ∈ µR(X) (ii) The union of elements of any sub collection of µR(X) is in µR(X). (iii) The intersection of the elements of any finite sub collection of µR(X) is in µR(X). Then µR(X) is called micro topology on U with respect to X. The triplet (U, τR(X), µR(X)) is called micro topological spaces and the elements of µR(X) are called micro open sets and the complement of a micro open set is called a micro closed set. Definition 4. [4] The micro closure of a set A is denoted by Mic-cl(A) and is defined as Mic-cl(A) = ∩{B:B is micro closed and A ⊆ B}. The micro interior of a set A is denoted by Mic-int(A) and is defined as Mic-int(A) = ∪{B:B is micro open and A ⊇ B}. Definition 5. [7] The union of all micro pre-open sets which are contained in A is called the micro pre-interior of A and is denoted by Mic-Pint(A) or by Mic-PA∗. As the union of micro pre-open sets is micro pre-open, Mic-PA∗ is micro pre-open always. micro pre-open is denoted by Mic-PO(U) and micro pre-closed is denoted by Mic-PF(U). Definition 6. [7] The intersection of micro pre-closed sets containing a set A is called the micro pre-closure of A and is denoted by Mic-Pcl(A) or by Mic-PA∗. Definition 7. [4] Let (U, τR(x), µR(X)) be a micro topological space and A ⊆ U . Then A is called micro pre-open if A ⊆ Mic-int(Mic-cl(A)). S. Stanley Roshan et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3156-3166 3158 Definition 8. [2] A point x ∈ X is said to be limit point of A if every neighborhood of x intersects A in some point other than x itself. Definition 9. [2] The set of all limit points of A is called the derived set of A and it denoted by D(A). Definition 10. [7] A point x ∈ U is said to be a micro pre-limit point of A iff for each U ∈ Mic-PO(U), U ∩ (A− {x}) ̸= ∅. Definition 11. [7] The set of all micro pre-limit points of A is said to be the micro pre-derived set of A and in denoted by Mic-PD(A). 2. micro pre-frontier In this section, we define and study the notions of micro pre-frontier and obtain its basic properties. Definition 12. micro pre-frontier of A ⊂ U is defined as Mic-PA∗− Mic-PA∗ and is denoted by Mic-Pfr(A). It is obvious that Mic-Pfr(A) ⊆ Mic-fr(A), the micro frontier of A. But in general the converse may not be true. micro pre-interior(A) is denoted as Mic-PA∗ and micro pre-closure is denoted as Mic-PA∗. Example 1. Let U = {a, b, c, d}, U\R = {{a, c}, {b, d}}, X = {a, c}, τR(X) = {U, ∅, {a, c}}, µ = {b} and µR(X) = {U, ∅, {b}, {a, c}, {a, b, c}}. If A = {b,c} then, Mic-cl(A) = {U} and Mic-int(A) = {b}, Mic-Pcl(A) = {b,c,d} and Mic-Pint(A) = {b,c}, where Mic-fr(A) = {a,c,d} and Mic-Pfr(A) = {d}. This shows that Mic-fr(A) ̸⊂ Mic-Pfr(A). Lemma 1. For a subset A of a space U, (i) Mic-PA∗ = Mic-PA∗ ∪ Mic-Pfr(A). (ii) Mic-PA∗ ∩ Mic-Pfr(A) = ∅ and (iii) Mic-Pfr(A) = Mic-PA∗ ∩ Mic-P(U −A)∗. Proof: By definition of Mic-Pfr(A), we have (i) Mic-PA∗ ∪ Mic-Pfr(A) = Mic-PA∗ ∪ (Mic-PA∗− Mic-PA∗) = Mic-PA∗. (ii) Mic-PA∗ ∩ Mic-Pfr(A) = Mic-PA∗ ∩ (Mic-PA∗− Mic-PA∗) = ∅. (iii) Mic-Pfr(A) = Mic-PA∗−Mic-PA∗ = Mic-PA∗ ∩ (U−Mic-PA∗) = Mic-PA∗ ∩ Mic- P(U −A)∗ by lemma 3.8(1) [7]. Lemma 2. Mic-Pfr(A) is micro pre-closed. Proof: By Lemma 1, Mic-Pfr(A) = Mic-PA∗ ∩ Mic-P(U−A)∗, which is micro pre-closed by corollary 3.9 [7]. S. Stanley Roshan et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3156-3166 3159 Definition 13. A subset A ⊂ U is called micro pre-regular if it is both micro pre-open and micro pre-closed set. The family of all micro pre-regular sets of U is denoted by Mic-PR(U). micro pre-closed is denoted by Mic-PF(U). Theorem 1. Mic-Pfr(A) = ∅ iff A ∈ Mic-PR(U). Proof: Let A ∈ Mic-PR(U). Then A ∈ Mic-PO(U) and A ∈ Mic-PF(U). Now, using results of Lemma 3.7 [7] and Theorem 3.16 [7] it follows that Mic-Pfr(A) = ∅. Conversely, let Mic-Pfr(A) = ∅. Then we show that A ∈ Mic-PR(U). Since by hypothesis, Mic-PA∗ − Mic-PA∗ = ∅. We have Mic-PA∗ = Mic-PA∗. But, Mic-PA∗ ⊂ A ⊂ Mic-PA∗. Therefore, it follows that A = Mic-PA∗ = Mic-PA∗ which means A ∈ Mic-PR(U). Theorem 2. Let A be subset of U. Then, the following holds. (i) Mic-Pfr(A) = Mic-Pfr(U −A). (ii) A ∈ Mic-PO(U) iff Mic-Pfr(A) ⊆ U −A. i.e., A ∩ Mic-Pfr(A) = ∅. (iii) A ∈ Mic-PF(U) iff Mic-Pfr(A) ⊆ A. Proof: (i) We have, Mic-Pfr(U −A) = (U−Mic-PA)∗ ∩ (U − (U−Mic-PA))∗ = (U−Mic-PA)∗ ∩ Mic-PA∗ = Mic-Pfr(A) by Lemma 1(3). (ii) Assume A ∈ Mic-PO(U). By definition, we have Mic-Pfr(A) = Mic-PA∗−Mic-PA∗ = Mic-PA∗ − A. Since A ∈ Mic-PO(U). Then, A ∩ Mic-Pfr(A) = A ∩ (Mic- PA∗ − A) = Mic-PA∗ ∩ (U − A) ∩ A = ∅. Conversely, if A ∩ Mic-Pfr(A) = ∅. Then, A ∩ Mic-PA∗ ∩ (U−Mic-PA∗) = ∅ implies A ∩ (U − Mic-PA∗) = ∅ as A ⊂ U − (U−Mic-PA∗) = Mic-PA∗, but on the other hand Mic-PA∗ ⊂ A. It follows that A = Mic-PA∗, which implies A ∈ Mic-PO(U). (iii) Assume A ∈ Mic-PF(U). Then, we have U − A ∈ Mic-PO(U). Then by (2), Mic- Pfr(U − A) ∩ (U − A) = ∅. But, by (1), Mic-Pfr (U − A) = Mic-Pfr(A). Hence Mic-Pfr(A) ∩ (U − A) = ∅. This shows that Mic-Pfr(A) ⊂ A. Conversely, if Mic- Pfr(A) ⊂ A, then Mic-PA∗ − Mic-PA∗ ⊂ A, which implies Mic-PA∗ ∪ (Mic-PA∗ − Mic-PA∗) ⊂ A ∪ Mic-PA∗ = A, which implies Mic-PA∗ ⊂ A by Lemma 1(1). But A ⊂ Mic-PA∗. It follows that A = Mic-PA∗. Hence A ∈ Mic-PF(U). Remark 1. Let A and B be subsets of space U. Then A ⊂ B does not imply that either Mic- Pfr(A) ⊂ Mic-Pfr(B) or Mic-Pfr(B) ⊂ Mic-Pfr(A). This can be verified by the following. Example 2. Let U = {a,b,c,d}, U/R = {{a,b}, {c,d}}, X = {b,c}, τR(X) = {U, ∅, {b, c}}, µ = b and µR(x) = {U, ∅, {b}, {b, c}}. Mic-PO(U) = {U, ∅, {b}, {a, b}, {b, c}, {b, d}, {a, b, c}, {b, c, d}, {a, b, d}}. Then, Case(1): Take A = {a} and B = {a,c}. Then A ⊂ B. Also Mic-PA∗ = {a}, Mic-PA∗ = {∅} and Mic-Pfr(A) = {a}. Mic-PB∗ = {a,c}, Mic-PB∗ = {∅} and Mic-Pfr(B) = {a,c}. This shows that Mic-Pfr(A) ⊂ Mic-Pfr(B). S. Stanley Roshan et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3156-3166 3160 Case(2): Take A = {a} and B = {a,c,d}. Also Mic-PA∗ = {a}, Mic-PA∗ = {∅} and Mic-Pfr(A) = {a}. Let Mic-PB∗ = {c,d,a}, Mic-PB∗ = {∅} and Mic-Pfr(B) = {a,c,d}. This shows that Mic-Pfr(A) ⊂ Mic-Pfr(B), where Mic-Pfr(B) ̸⊂ Mic-Pfr(A). Theorem 3. If A ∈ Mic-PO(U) ∪ Mic-PF(U), then Mic-Pfr(A) = Mic-Pfr(Mic-Pfr(A)). Proof: It follows by Lemma 1(3), Lemma 2 and Theorem 2 (2, 3). Corollary 1. For every A ⊂ U, Mic-Pfr(Mic-Pfr(Mic-Pfr(A))) = Mic-Pfr(Mic-Pfr(A)). Proof: It is obvious. Lemma 3. A subset A of U is micro pre-closed iff A = Mic-Pcl(A). Theorem 4. For a subset A of space U, the following statements hold (i) A is Mic-PO iff Mic-Pfr(A) = Mic-PD(A). (ii) Mic-Pfr(Mic-Pfr(A)) ⊆ Mic-Pfr(A). (iii) Mic-Pfr(Mic-Pcl(A)) ⊆ Mic-Pfr(A). (iv) Mic-Pint(A) = A − Mic-Pfr(A) Proof: (i) Let A be micro pre-open then Mic-Pint(A) = A. Since Mic-Pfr(A) = Mic-Pcl(A) − Mic-Pint(A) = Mic-Pcl(A)−A. By Lemma 4.9 [7] we have Mic-Pcl(A) = A ∪ Mic-PD(A). Therefore Mic-Pfr(A) = [A ∪ Mic-PD(A)]− A = Mic-PD(A). Conversely, Let Mic-Pfr(A) = Mic-PD(A). i.e., Mic-Pcl(A) − Mic-Pint(A) = [A ∪ Mic-PD(A)] - Mic-Pint(A) = Mic-PD(A) ⇒ A − Mic-Pint(A) = ∅ implies A ⊂ Mic-Pint(A) −→ (1) and Mic-Pint(A) ⊂ A −→ (2). Therefore from (1) and (2) we have Mic-Pint(A) = A is micro pre-open. (ii) Now Mic-Pfr(Mic-Pfr(A)) ⊆ Mic-Pcl(Mic-Pfr(A)) ∩ Mic-Pcl(U − Mic-Pfr(A)) ⇒ Mic-Pfr(Mic-Pfr(A)) ⊆ Mic-Pcl(Mic-Pfr(A)) ⊆ Mic-Pfr(A). (iii) Mic-Pfr(Mic-Pcl(A)) ⊆ Mic-Pcl(Mic-Pcl(A)) − Mic-Pint(Mic-Pcl(A)) ⊆ Mic-Pcl(A) − Mic-Pint(A) ⊆ Mic-Pfr(A) (iv) It is obvious from the definition of micro pre-interior and micro pre-frontier. 3. micro pre-exterior In this section, we define and study the notions of micro pre-exterior and obtain its basic properties. Definition 14. A point x ∈ U is called micro pre-exterior point of a subset A of U if x is micro pre-interior point of (U − A) and set of all micro pre-exterior points of A is called micro pre-exterior of A and denoted by Mic-Pext(A). Therefore Mic-Pext(A) = Mic-Pint(U −A). S. Stanley Roshan et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3156-3166 3161 Theorem 5. For a subset A of a space U the following statements hold (i) Mic-ext(A) ⊆ Mic-Pext(A). (ii) Mic-Pext(A) ⊂ Mic-PO(U). (iii) Mic-Pext(A) = U−Mic-Pcl(A). (iv) Mic-Pext(Mic-Pext(A)) = Mic-Pint(Mic-Pcl(A)). (v) If A ⊂ B then Mic-Pext(B) ⊆ Mic-Pext(A). (vi) Mic-Pext(A ∪ B) ⊆ Mic-Pext(A) ∪ Mic-Pext(B). (vii) Mic-Pext(A) ∩ Mic-Pext(B) ⊆ Mic-Pext(A ∩ B). (viii) Mic-Pext(U) = ∅ and Mic-Pext(∅) = U. (ix) Mic-Pext(A) = Mic-pext[U − Mic-Pext(A)]. (x) Mic-Pint(A) ⊆ Mic-Pext[Mic-Pext(A)]. (xi) Mic-Pint(A), Mic-Pext(A) and Mic-Pfr(A) are mutually disjoint and U = Mic- Pint(A) ∪ Mic-Pext(A) ∪ Mic-Pfr(A). (xii) A ∩ Mic-Pext(A) = ∅. Proof: (i) Let x ∈ Mic-ext(A) ⇒ x ∈ Mic-int(U − A). There exists G ∈ µR(x) such that x ∈ G ⊆ (U −A). Also G ∈ Mic-PO(U, X). Therefore x ∈ G ⊆ (U −A) for Mic-PO set G ⇒ (U − A) is Mic-Pint of x, x ∈ Mic-Pint(U − A)i.e., x ∈ Mic-Pext(A). Hence Mic-ext(A) ⊆ Mic-Pext(A). (ii) Now Mic-Pint[Mic-Pext(A)] = Mic-Pint[Mic-Pint(U − A)] = Mic-Pint(U − A) = Mic-Pext(A) ⇒ is contained in Mic-PO(U). (iii) Mic-Pext(A) = Mic-Pint(U −A) = U−Mic-Pcl(A). (iv) Mic-Pext[Mic-Pext(A)] = Mic-Pext[U−Mic-cl(A)] by (3), Mic-Pext[U−Mic-cl(A)] = Mic-Pint[U−[U−Mic-Pcl(A)]] = Mic-Pint[Mic-Pcl(A)]. (v) If A ⊂ B then (U − B) ⊂ (U − A) ⇒ Mic-Pint(U − B) ⊆ Mic-Pint(U − A) i.e., Mic-Pext(B) ⊆ Mic-Pext(A). (vi) Since A ⊂ (A ∪ B) and B ⊂ (A ∪ B) ⇒ Mic-Pext(A ∪ B) ⊆ Mic-Pext(A) ∪ Mic-Pext(A ∪ B) ⊆ Mic-Pext(B). Therefore Mic-Pext(A ∪ B) ⊆ Mic-Pext(A) ∪ Mic-Pext(B). S. Stanley Roshan et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3156-3166 3162 (vii) We have (A ∩ B) ⊂ A, (A ∩ B) ⊂ B. ⇒ Mic-Pext(A) ⊆ Mic-Pext(A ∩ B) and Mic-Pext(B) ⊆ Mic-Pext(A ∩ B) ⇒ Mic- Pext(A) ∩ Mic-Pext(B) ⊆ Mic-Pext(A ∩ B). (viii) Mic-Pext(U) = Mic-Pint(U − U) = Mic-Pint(∅) = ∅ and Mic-Pext(∅) = Mic- Pint(U−∅) = Mic-Pint(U) = U. (ix) Mic-Pext[U − Mic-Pext(A)] = Mic-Pint[U−(U− Mic-Pint(A))] = Mic-Pint(Mic- Pext((A)) = Mic-Pint[Mic-Pint(U −A)] = Mic-Pint(U −A) = Mic-Pext(A). (x) By the definition Mic-Pext(A) ⊂ (U − A) then from (5) Mic-Pext(U − A) ⊂ Mic- Pext[Mic-Pext(A)] i.e., Mic-Pint(A) ⊂ Mic-Pext[Mic-Pext(A)]. (xi) Let us assume that Mic-Pext(A) ∩ Mic-Pint(A) ̸= ∅ therefore there exists x ∈ Mic- Pext(A) ∩ Mic-Pint(A) ⇒ x ∈ Mic-Pext(A) and x ∈ Mic-Pint(A) ⇒ x ∈ (U − A) and x ∈ A which is not possible. Therefore our assumption is wrong. Hence Mic- Pext(A) ∩ Mic-Pint(A) = ∅ similarly other two results. We have Mic-Pext(A) = U− Mic-cl(A) =U−[Mic-Pint(A) ∪ Mic-Pfr(A)] that implies U = Mic-Pint(A) ∪ Mic-Pext(A) ∪ Mic-Pfr(A). (xii) Obvious. In general, the converse of (6) and (7) are not true i.e., Mic-Pext(A) ∪ Mic-Pext(B) ̸⊂ Mic-Pext(A ∪ B) and Mic-Pext(A ∩ B) ̸⊂ Mic-Pext(A) ∩ Mic-Pext(B). Example 3. Let U = {a, b, c, d}, U\R = {{a, b}, {c, d}}, X = {b, c}, τR(X) = {U, ∅, {b, c}}, µ = {b, d} and µR(X) = {U, ∅, {b}, {b, c}, {b, d}, {b, c, d}}. Mic-PO(U) = {U, ∅, {b}, {a, b}, {b, c}, {b, d}, {a, b, c}, {b, c, d}, {a, b, d}}. Let A = {c,d,a} and B = {c,d}. Then Mic-Pext(A) = {b} and Mic-Pext(B) = {a,b}. Mic-Pext(A ∪ B) = {b}. ⇒ Mic-Pext(A) ∪ Mic-Pext(B) ̸⊂ Mic-Pext(A ∪ B) and Mic-Pext(A ∩ B) = {a,b} which implies Mic-Pext(A ∩ B) ̸⊂ Mic-Pext(A) ∩ Mic-Pext(B). Theorem 6. Mic-Pfr(A) ∩ Mic-Pext(A) = ∅. Proof: Let x ∈ Mic-Pfr(A) i.e., x ∈ (Mic-Pcl(A) − Mic-Pint(A)). If x ∈ Mic-Pcl(A) then x /∈ Mic-Pint(A). We know that Mic-Pcl(A) ∩ Mic-Pint(U − A) = ∅. Therefore x /∈ Mic-Pint(U − A) implies x /∈ Mic-Pext(A). Hence Mic-Pfr(A) ∩ Mic-Pext(A) = ∅. 4. micro pre-border In this section, we define and study the notions of micro pre-border and obtain its basic properties. Definition 15. Let A be a subset of a space U. Then the micro pre-border of A is defined as Mic-Pbr(A) = A − Mic-Pint(A). Theorem 7. For a subset of U, the following statements holds. S. Stanley Roshan et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3156-3166 3163 (i) Mic-Pbr(A) ⊆ Mic-br(A) where br(A) denote the border of A. (ii) A = Mic-Pint(A) ∪ Mic-Pbr(A). (iii) Mic-Pint(A) ∩ Mic-Pbr(A) = ∅. (iv) If A is Mic-PO then Mic-Pbr(A) = ∅. (v) Mic-Pint(Mic-Pbr(A)) = ∅. (vi) Mic-Pbr(Mic-Pbr(A)) = Mic-Pbr(A). (vii) Mic-Pbr(A) = A ∩ Mic-Pcl(U −A). Proof: (i) Obvious from the definitions of micro pre-border and micro border of A. (ii) Obvious from the definitions of micro pre-border of A. (iii) Obvious from the definitions of micro pre-border of A. (iv) If A is Mic-PO, then A = Mic-Pint(A). Hence the result follows. (v) If x ∈ Mic-Pint(Mic-Pbr(A)), then x ∈ Mic-Pbr(A). Now, Mic-Pbr(A) ⊂ A implies Mic-Pint(Mic-Pbr(A)) ⊂ Mic-Pint(A). Hence x ∈ Mic-Pint(A) which is a contra- diction to x ∈ Mic-Pbr(A). Thus Mic-Pint(Mic-Pbr(A)) = ∅. (vi) Mic-Pbr(Mic-Pbr(A)) = Mic-Pbr(A − Mic-Pint(A)) = (A − Mic-Pint(A))−Mic- Pint(A − Mic-Pint(A)) which is Mic-Pbr(A) − ∅, by (4). Hence, Mic-Pbr(Mic- Pbr(A)) = Mic-Pbr(A). (vii) Mic-Pbr(A) = A − Mic-Pint(A) =A−(U−(Mic-Pcl(U−A))) = A ∩ Mic-Pcl(U−A). Theorem 8. For a subset of U, the following condition hold. (i) Mic-Pbr(A) ⊆ Mic-Pfr(A). (ii) Mic-Pext(A) ∩ Mic-Pbr(A) = ∅. Proof: (i) Let, x ∈ Mic-Pbr(A) i.e., x ∈ A − Mic-Pint(A). By Theorem 3.16 [7] A = Mic-pint(A) if A is Mic-PO. If A is not Mic-PO then Mic-Pint(A) ⊂ A. Therefore in general Mic-Pint(A) ⊆ A. So x ∈ Mic-Pint(A). It is obvious that if x ∈ Mic-Pint(A) then x /∈ Mic-Pcl(A). Therefore x ∈ (Mic-Pcl(A) − Mic-Pint(A)) implies x ∈ Mic-Pfr(A). Hence Mic-Pbr(A) ⊆ Mic-Pfr(A). (ii) Let x ∈ Mic-Pext(A) i.e., x ∈ Mic-Pint(U−A) where x ∈ Mic-Pint(A). By Theorem 3.16 [7] A = Mic-pint(A) if A is Mic-PO. If A is not Mic-PO then Mic-Pint(A) ⊂ A. Therefore in general Mic-Pint(A) ⊆ A. Therefore x /∈ A − Mic-Pint(A) implies x /∈ Mic-Pbr(A). Hence Mic-Pext(A) ∩ Mic-Pbr(A) = ∅. S. Stanley Roshan et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3156-3166 3164 5. micro pre-kernel In this section, we define and study the notions of micro pre-kernel and obtain its basic properties. Definition 16. For any A ⊂ U, Mic-Pker(A) is defined as the intersection of all micro pre-open sets containing A. In notation, Mic-Pker(A) = ⋂ {M/A ⊂ M, M ∈ Mic-PO}. Lemma 4. For subsets A, B and Ai(i ∈ I, where I is an index set) of a micro topological space (U, µR(x)), the following holds. (i) A ⊆ Mic-Pker(A). (ii) If A ⊂ B, then Mic-Pker(A) ⊂ Mic-Pker(B). (iii) Mic-Pker(Mic-Pker(A)) = Mic-Pker(A). (iv) Mic-Pker( ⋃ Ai/ i ∈ I) ⊆ ⋃ {Mic-Pker (Ai)/ i ∈ I}. (v) Mic-Pker( ⋂ Ai/ i ∈ I) ⊆ ⋂ {Mic-Pker (Ai)/ i ∈ I}. Proof: (i) It follows by the definition of Mic-Pker(A). (ii) Suppose x /∈ Mic-Pker(B), then there exists a subset S ∈ Mic-PO such that B ⊂ S with x /∈ S. Since A ⊂ B, x /∈ Mic-Pker(A). Thus Mic-Pker(A) ⊂ Mic-Pker(B). (iii) Follows from (1) and definition of Mic-Pker(A). (iv) For each i ∈ I, Mic-Pker (Ai) ⊆ Mic-Pker( ⋃ i∈I Ai). Therefore we have ⋃ i∈I{Mic- Pker(Ai)} ⊆ Mic-Pker ( ⋃ i∈I Ai). (v) Suppose that x /∈ ⋂ {Mic-Pker(Ai/i ∈ I)} then there exists an i0 ∈ I, such that x /∈ Mic-Pker(Ai0) and there exists a micro pre-open set S such that x /∈ S and Ai0 ⊂ S. We have ⋂ i∈I Ai ⊆ Ai0 ⊆ S and x /∈ S. Therefore x /∈ Mic-Pker { ⋂ Ai/i ∈ I}. Hence Mic-Pker( ⋂ Ai/i ∈ I) ⊆ ⋂ Mic-Pker(Ai)/i ∈ I. Theorem 9. Let A and B be subsets of U, then the following conditions hold. (i) Mic-Pker(A) ⊆ Mic-ker(A). (ii) Mic-Pker(A) ∩ Mic-Pker(B) ⊂ Mic-Pker(A ∪ B). (iii) Mic-Pker(A ∩ B) ⊂ Mic-Pker(A) ∪ Mic-Pker(B). (iv) Mic-Pcl(A) ∩ Mic-Pker(A) = A. (v) Mic-Pker(A) ∩ Mic-Pfr(A) = Mic-Pbr(A). proof: REFERENCES 3165 (i) Let x ∈ Mic-Pker(A). ⇒ x ∈ ⋂ {M/A ⊂ M, M ∈ Mic-PO} ⇒ x ∈ ⋂ {M/A ⊂ M, M ∈ micro-Open(µR(x))} Since every micro open is micro pre-open, x ∈ Mic-ker(A). (ii) Let x ∈{Mic-Pker(A) ∩ Mic-Pker(B)} ⇒ x ∈ Mic-Pker(A) and x ∈ Mic-Pker(B) Therefore, x ∈ Mic-Pker(A ∪ B). (iii) Let x ∈ Mic-Pker(A ∩ B) ⇒ x ∈ Mic-Pker(A) and x ∈ Mic-Pker(B) Therefore, x ∈ Mic-Pker(A) ∪ Mic-Pker(B). (iv) Let x ∈ Mic-Pcl(A) ∩ Mic-Pker(A) By Lemma 3.7(1) [7] A ⊆ Mic-Pcl(A) and by (1) A ⊆ Mic-Pker(A). ⇒ x ∈ A ⊆ Mic-Pcl(A) and x ∈ A ⊆ Mic-Pker(A). Therefore, x ∈ A. (v) Let x ∈ Mic-Pker(A) ∩ Mic-Pfr(A). To prove, x ∈ Mic-Pbr(A) i.e, x ∈ A− Mic- Pint(A). Since Mic-Pker(A) = ⋂ {M/A ⊂ M, M ∈ Mic-PO} and Mic-Pfr(A) = Mic-Pcl(A)−Mic-Pint(A). ⇒ x ∈ ⋂ {M/A ⊂ M, M ∈ Mic-PO} ∩ Mic-Pcl(A)−Mic-Pint(A). By Lemma 3.7(1) [7] and Lemma 4(1), we have ⇒ x ∈ A ∩ (A − Mic-Pint(A)) ⇒ x ∈ A and x ∈ A − Mic-Pint(A) ⇒ x ∈ A and x ∈ Mic-Pbr(A) Therefore, x ∈ Mic-Pbr(A). 6. Conclusion In this paper, we introduced the notions of micro pre-frontier, micro pre-exterior, micro pre-border and micro pre-kernel by employing the concept of frontier, exterior, border and kernel elucidating various associated properties. Our intent is to further elaborate on these findings in forthcoming research endeavors, with a particular focus on exploring practical applications. References [1] Lellis Thivagar. 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