International Journal of Analysis and Applications Volume 18, Number 4 (2020), 572-585 URL: https://doi.org/10.28924/2291-8639 DOI: 10.28924/2291-8639-18-2020-572 MICRO SEPARATION AXIOMS HARIWAN Z. IBRAHIM∗ Department of Mathematics, Faculty of Education, University of Zakho, Zakho, Kurdistan Region, Iraq ∗Corresponding author: hariwan.ibrahim@uoz.edu.krd Abstract. In this paper, some new types of spaces are defined and studied in micro topological spaces namely, Micro T0, Micro T1, Micro T2, Micro R0 and Micro R1 spaces. Properties and the relationships of these spaces are introduced. Finally, the relationships between these spaces and the related concepts are investigated. 1. Introduction Topology and its branches have become hot topics not only for almost all fields of mathematics, but also for many areas of science such as chemistry [21], and information systems [23]. The notion of rough sets was introduced by Pawlak [22]. Rough set theory is an important tool for data mining. Lower and upper approximation operators are two important basic concepts in the rough set theory. The classical Pawlak rough approximation operators are based on equivalence relations and have been extended to relation-based generalized rough approximation operators. The notation of nano topology was introduced by Thivagar et al [25, 26] which was defined in terms of approximations and boundary region of a subset of an universe using an equivalence relation on it. The concept of micro topology was introduced and investigated by Chandrasekar [1]. The notion of micro T1 2 space was introduced by Ibrahim [16]. In the past few years, different forms of separation axioms have been studied [2–14, 17–20]. Received February 21st, 2020; accepted March 19th, 2020; published May 11th, 2020. 2010 Mathematics Subject Classification. Primary 22A05, 22A10; Secondary 54C05. Key words and phrases. micro topology; micro open set; micro T0; micro T1; micro R0; micro R1. ©2020 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 572 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-18-2020-572 Int. J. Anal. Appl. 18 (4) (2020) 573 2. Preliminaries The following recalls requisite ideas and preliminaries necessitated in the sequel of this work. Definition 2.1. [24] Let U be a nonempty 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. (1) The lower approximation of U with respect to R is the set of all objects, which can be for certain classified as X with respect to R and its is denoted by LR(X). That is, LR(X) = ⋃ x∈U{R(x) : R(x) ⊆ X}, where R(x) denotes the equivalence class determined by x. (2) The upper approximation of U 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 LR(X). That is, LR(X) = ⋃ x∈U{R(x) : R(x)∩X 6= φ}. (3) The boundary region of U with respect to R is the set of all objects, which can be classified neither as X nor as not-X with respect to R and it is denoted by BR(X). That is, BR(X) = LR(X)−LR(X). Definition 2.2. [25, 26] Let U be the universe, R be an equivalence relation on U and τR(X) = {U,φ,LR(X),LR(X),BR(X)}, where X ⊆ U. Then, τR(X) satisfies the following axioms: (1) U and φ ∈ τR(X). (2) The union of the elements of any subcollection of τR(X) is in τR(X). (3) The intersection of the elements of any finite subcollection of τR(X) is in τR(X). That is, τR(X) is a topology on U called the nano topology on U with respect to X. We call (U,τR(X)) as the nano topological space. The elements of τR(X) are called as nano open sets. A subset F of U is nano closed if its complement is nano open. Definition 2.3. [1] Let (U,τR(X)) be a nano topological space. Then, µR(X) = {N ∪ (N ′ ∩µ) : N,N ′ ∈ τR(X) and µ /∈ τR(X)} is called the micro topology on U with respect to X. The triplet (U,τR(X),µR(X)) is called micro topological space 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 2.4. [1] 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}. Definition 2.5. [1] Let (U,τR(X),µR(X)) be a micro topological space. Let A and B be any two subsets of U. Then: (1) A is a Micro closed set if and only if Mic-cl(A) = A. (2) If A ⊆ B, then Mic-cl(A) ⊆ Mic-cl(B). (3) Mic-cl(Mic-cl(A)) = Mic-cl(A). Int. J. Anal. Appl. 18 (4) (2020) 574 Remark 2.1. [15] Let (U,τR(X),µR(X)) be a micro topological space and A be any subset of U. Then: (1) Mic-cl(A) is Micro closed. (2) A ⊆ Mic-cl(A). (3) x ∈ Mic-cl(A) if and only if for every Micro open subset L of U containing x, A∩L 6= φ. Definition 2.6. [16] Let (U,τR(X),µR(X)) be a micro topological space. A subset A of U is said to be a Micro generalized closed (briefly, Micro g.closed) if Mic-cl(A) ⊆ L whenever A ⊆ L and L is a Micro open set in U. Definition 2.7. [16] Let (U,τR(X),µR(X)) be a micro topological space. Then, a subset A of U is called a Micro Difference set (briefly, MD-set) if there are L,K ∈ µR(X) such that L 6= U and A = L\K. Definition 2.8. [16] Let (U,τR(X),µR(X)) be a micro topological space and A be a subset of U. Then, the Micro kernel of A denoted by Mker(A) is defined to be the set Mker(A) = ∩{L ∈ µR(X): A ⊆ L}. Definition 2.9. [16] Let (U,τR(X),µR(X)) be a micro topological space. Then, U is said to be Micro T1 2 if every Micro g.closed in U is Micro closed. Definition 2.10. [16] Let (U,τR(X),µR(X)) be a micro topological space. Then, U is said to be Micro symmetric if for x and y in U such that x ∈ Mic-cl({y}) implies y ∈ Mic-cl({x}). Definition 2.11. [16] Let (U,τR(X),µR(X)) be a micro topological space. Then, U is said to be: (1) Micro D0 if for any pair of distinct points x and y of U there exists a MD-set of U containing x but not y or a MD-set of U containing y but not x. (2) Micro D1 if for any pair of distinct points x and y of U there exists a MD-set of U containing x but not y and a MD-set of U containing y but not x. (3) Micro D2 if for any pair of distinct points x and y of U there exist disjoint MD-set G and E of U containing x and y, respectively. Theorem 2.1. [16] Let (U,τR(X),µR(X)) be a micro topological space. Then, U is a Micro T1 2 if and only if {x} is Micro closed or Micro open, for each x ∈ U. Remark 2.2. [16] Let (U,τR(X),µR(X)) be a micro topological space. If U is Micro Dk, then it is Micro Dk−1, for k = 1, 2. Theorem 2.2. [16] Let (U,τR(X),µR(X)) be a micro topological space. Then, then the following statements are equivalent: Int. J. Anal. Appl. 18 (4) (2020) 575 (1) U is a Micro symmetric. (2) {x} is Micro g.closed, for each x ∈ U. Theorem 2.3. [16] Let (U,τR(X),µR(X)) be a micro topological space and x ∈ U. Then, y ∈ Mker({x}) if and only if x ∈ Mic-cl({y}). Theorem 2.4. [16] Let (U,τR(X),µR(X)) be a micro topological space and A be a subset of U. Then, Mker(A) = {x ∈ U: Mic-cl({x}) ∩A 6= φ}. Theorem 2.5. [16] Let (U,τR(X),µR(X)) be a micro topological space. Then, for any points x and y in U the following statements are equivalent: (1) Mker({x}) 6= Mker({y}). (2) Mic-cl({x}) 6= Mic-cl({y}). Definition 2.12. [1] Let (U,τR(X),µR(X)) and (V,τR(Y ),µR(Y )) be two micro topological spaces. Then, A function f : U → V is said to be: Micro-continuous if f−1(K) is Micro open in U, for every Micro open set K in V . 3. Micro Tk (k = 0, 1, 2) The following definitions are introduced via Micro open sets. Definition 3.1. Let (U,τR(X),µR(X)) be a micro topological space. Then, U is said to be: (1) Micro T0 if for each pair of distinct points x,y in U, there exists a Micro open set L such that either x ∈ L and y /∈ L or x /∈ L and y ∈ L. (2) Micro T1 if for each pair of distinct points x,y in U, there exist two Micro open sets L and K such that x ∈ L but y /∈ L and y ∈ K but x /∈ K. (3) Micro T2 if for each distinct points x,y in U, there exist two disjoint Micro open sets L and K containing x and y respectively. Theorem 3.1. Let (U,τR(X),µR(X)) be a micro topological space. Then, U is Micro T0 if and only if for each pair of distinct points x,y of U, Mic-cl({x}) 6= Mic-cl({y}). Proof. Necessity. Let U be Micro T0 and x,y be any two distinct points of U. Then, there exists a Micro open set L containing x or y, say x but not y. Then, U \L is a Micro closed set which does not contain x but contains y. Since Mic-cl({y}) is the smallest Micro closed set containing y, then Mic-cl({y}) ⊆ U \L and therefore x /∈ Mic-cl({y}). Consequently Mic-cl({x}) 6= Mic-cl({y}). Sufficiency. Suppose that x,y ∈ U, x 6= y and Mic-cl({x}) 6= Mic-cl({y}). Let z be a point of U such that z ∈ Mic-cl({x}) but z /∈ Mic-cl({y}). We claim that x /∈ Mic-cl({y}). For, if x ∈ Mic-cl({y}) then Int. J. Anal. Appl. 18 (4) (2020) 576 Mic-cl({x}) ⊆ Mic-cl({y}). This contradicts the fact that z /∈ Mic-cl({y}). Consequently x belongs to the Micro open set U \Mic-cl({y}) to which y does not belong. � Theorem 3.2. Let (U,τR(X),µR(X)) be a micro topological space. Then, U is Micro T1 if and only if the singletons are Micro closed sets. Proof. Let U be Micro T1 and x any point of U. Suppose y ∈ U\{x}, then x 6= y and so there exists a Micro open set L such that y ∈ L but x /∈ L. Consequently y ∈ L ⊆ U \{x}, that is U \{x} = ∪{L : y ∈ U \{x}} which is Micro open. Conversely, suppose {p} is Micro closed for every p ∈ U. Let x,y ∈ U with x 6= y. Now, x 6= y implies y ∈ U \{x}. Hence, U \{x} is a Micro open set contains y but not x. Similarly U \{y} is a Micro open set contains x but not y. Accordingly U is Micro T1. � Theorem 3.3. Let (U,τR(X),µR(X)) be a micro topological space. Then, the following statements are equivalent: (1) U is Micro T2. (2) Let x ∈ U. For each y 6= x, there exists a Micro open set L containing x such that y /∈ Mic-cl(L). (3) For each x ∈ U, ∩{Mic-cl(L) : L ∈ µR(X) and x ∈ L} = {x}. Proof. (1) ⇒ (2): Since U is Micro T2, then there exist disjoint Micro open sets L and K containing x and y respectively. So, L ⊆ U \K. Therefore, Mic-cl(L) ⊆ U \K. So, y /∈ Mic-cl(L). (2) ⇒ (3): If possible for some y 6= x, we have y ∈ Mic-cl(L) for every Micro open set L containing x, which then contradicts (2). (3) ⇒ (1): Let x,y ∈ U and x 6= y. Then, there exists a Micro open set L containing x such that y /∈ Mic- cl(L). Let K = U \Mic-cl(L), then y ∈ K, x ∈ L and L∩K = φ. Thus, U is Micro T2. � Theorem 3.4. Let (U,τR(X),µR(X)) be a micro topological space. Then, then the following statements are hold: (1) Every Micro T2 space is Micro T1. (2) Every Micro T1 space is Micro T1 2 . (3) Every Micro T1 2 space is Micro T0. Proof. (1) The proof is straightforward from the definitions. (2) The proof is obvious by Theorem 3.2. (3) Let x and y be any two distinct points of U. By Theorem 2.1, the singleton set {x} is Micro closed or Micro open. (a) If {x} is Micro closed, then U \{x} is Micro open. So y ∈ U \{x} and x /∈ U \{x}. Therefore, we have U is Micro T0. Int. J. Anal. Appl. 18 (4) (2020) 577 (b) If {x} is Micro open, then x ∈{x} and y /∈{x}. Therefore, we have U is Micro T0. � Remark 3.1. Let (U,τR(X),µR(X)) be a micro topological space. Then, (1) if U is Micro T1, then µR(X) is discrete micro topology on U. (2) U is Micro T1 if and only if it is Micro T2. Example 3.1. Consider U = {a,b,c} with U/R = {{a},{b,c}} and X = {a}. Then, τR(X) = {U,φ,{a}}. If µ = {a,c}, then µR(X) = {U,φ,{a},{a,c}}. Then, U is Micro T0 but not Micro T1 2 . Example 3.2. Consider U = {a,b,c} with U/R = {{c},{a,b}} and X = {a,b}. Then, τR(X) = {U,φ,{a,b}}. If µ = {a,c}, then µR(X) = {U,φ,{a},{a,b},{a,c}}. Then, U is Micro T1 2 but not Mi- cro T1. Remark 3.2. Let (U,τR(X),µR(X)) be a micro topological space. If U is Micro Tk, then it is Micro Dk, for k = 0, 1, 2. Proof. Obvious. � Theorem 3.5. Let (U,τR(X),µR(X)) be a micro topological space. Then, U is Micro D0 if and only if it is Micro T0. Proof. Suppose that U is Micro D0. Then, for each distinct pair x,y ∈ U, at least one of x,y, say x, belongs to a MD-set G but y /∈ G. Let G = L1 \ L2 where L1 6= U and L1,L2 ∈ µR(X). Then, x ∈ L1, and for y /∈ G we have two cases: (a) y /∈ L1 (b) y ∈ L1 and y ∈ L2. In case (a), x ∈ L1 but y /∈ L1. In case (b), y ∈ L2 but x /∈ L2. Thus in both the cases, we obtain that U is Micro T0. Conversely, if U is Micro T0, by Remark 3.2, U is Micro D0. � Corollary 3.1. If U is Micro D1, then it is Micro T0. Proof. Follows from Remark 2.2 and Theorem 3.5. � Here is an example which shows that the converse of Corollary 3.1 is not true in general. Example 3.3. Consider U = {a,b,c} with U/R = {{a},{b,c}} and X = {a}. Then, τR(X) = {U,φ,{a}}. If µ = {a,b}, then µR(X) = {U,φ,{a},{a,b}}. Then, U is Micro T0 but not Micro D1 because there is no MD-set containing c but not b. Int. J. Anal. Appl. 18 (4) (2020) 578 Corollary 3.2. Let (U,τR(X),µR(X)) be a micro topological space. If U is Micro T1, then it is Micro symmetric. Proof. In Micro T1, every singleton is Micro closed and therefore is Micro g.closed. Then, by Theorem 2.2, U is Micro symmetric. � Corollary 3.3. Let (U,τR(X),µR(X)) be a micro topological space. Then, the following statements are equivalent: (1) U is Micro symmetric and Micro T0. (2) U is Micro T1. Proof. By Corollary 3.2 and Theorem 3.4, it suffices to prove only (1) ⇒ (2). Let x 6= y and as U is Micro T0, we may assume that x ∈ L ⊆ U \ {y} for some L ∈ µR(X). Then, x /∈ Mic-cl({y}) and hence y /∈ Mic-cl({x}). There exists a Micro open set K such that y ∈ K ⊆ U \{x} and thus U is a Micro T1 space. � Theorem 3.6. Let (U,τR(X),µR(X)) be a micro topological space. If U is Micro symmetric, then the following statements are equivalent: (1) U is Micro T0. (2) U is Micro T1 2 . (3) U is Micro T1. Proof. (1) ⇔ (3): Obvious from Corollary 3.3. (3) ⇒ (2) and (2) ⇒ (1): Directly from Theorem 3.4. � Corollary 3.4. Let (U,τR(X),µR(X)) be a micro topological space. If U is Micro symmetric, then the following statements are equivalent: (1) U is Micro T0. (2) U is Micro D1. (3) U is Micro T1. Proof. (1) ⇒ (3). Follows from Corollary 3.3. (3) ⇒ (2) ⇒ (1). Follows from Remark 3.2 and Corollary 3.1. � Definition 3.2. A function f : U → V is called Micro-open if the image of every Micro open set in U is a Micro open set in V . Theorem 3.7. Suppose that f : U → V is Micro-open and surjective. Then: Int. J. Anal. Appl. 18 (4) (2020) 579 (1) If U is Micro T0, then V is Micro T0. (2) If U is Micro T1, then V is Micro T1. (3) If U is Micro T2, then V is Micro T2. Proof. We prove only the case for Micro T1 the others are similarly. Let U be Micro T1 and y1,y2 ∈ V with y1 6= y2. Since f is surjective, so there exist distinct points x1,x2 of U such that f(x1) = y1 and f(x2) = y2. Since U is Micro T1, then there exist Micro open sets G and H such that x1 ∈ G but x2 /∈ G and x2 ∈ H but x1 /∈ H. Since f is Micro-open, then f(G) and f(H) are Micro open sets of V such that y1 = f(x1) ∈ f(G) but y2 = f(x2) /∈ f(G), and y2 = f(x2) ∈ f(H) but y1 = f(x1) /∈ f(H). Hence, V is Micro T1. � Theorem 3.8. If f : U → V is a Micro-continuous injective function and V is Micro T2, then U is Micro T2. Proof. Let x and y in U be any pair of distinct points, then there exist disjoint Micro open sets A and B in V such that f(x) ∈ A and f(y) ∈ B. Since f is Micro-continuous, then f−1(A) and f−1(B) are Micro open in U containing x and y respectively, we have f−1(A) ∩f−1(B) = φ. Thus, U is Micro T2. � 4. Micro Rk (k = 0, 1) Definition 4.1. Let (U,τR(X),µR(X)) be a micro topological space. Then, U is said to be Micro R0 if L is a Micro open set and x ∈ L, then Mic-cl({x}) ⊆ L. Theorem 4.1. Let (U,τR(X),µR(X)) be a micro topological space. Then, the following properties are equivalent: (1) U is Micro R0. (2) For any F ∈ µcR(X), x /∈ F implies F ⊆ L and x /∈ L for some L ∈ µR(X). Where µ c R(X) is the family of all Micro closed sets. (3) For any F ∈ µcR(X), x /∈ F implies F ∩Mic-cl({x}) = φ. (4) For any distinct points x and y of U, either Mic-cl({x}) = Mic-cl({y}) or Mic-cl({x}) ∩ Mic- cl({y}) = φ. Proof. (1) ⇒ (2): Let F ∈ µcR(X) and x /∈ F . Then by (1), Mic-cl({x}) ⊆ U \F. Set L = U \Mic-cl({x}), then L is a Micro open set such that F ⊆ L and x /∈ L. (2) ⇒ (3): Let F ∈ µcR(X) and x /∈ F . Then, there exists L ∈ µR(X) such that F ⊆ L and x /∈ L. Since L ∈ µR(X), then L∩Mic-cl({x}) = φ and F ∩Mic-cl({x}) = φ. (3) ⇒ (4): Suppose that Mic-cl({x}) 6= Mic-cl({y}) for distinct points x,y ∈ U. Then, there exists z ∈ Mic- cl({x}) such that z /∈ Mic-cl({y}) (or z ∈ Mic-cl({y}) such that z /∈ Mic-cl({x})). There exists K ∈ µR(X) Int. J. Anal. Appl. 18 (4) (2020) 580 such that y /∈ K and z ∈ K; hence x ∈ K. Therefore, we have x /∈ Mic-cl({y}). By (3), we obtain Mic- cl({x}) ∩Mic-cl({y}) = φ. (4) ⇒ (1): Let K ∈ µR(X) and x ∈ K. For each y /∈ K, x 6= y and x /∈ Mic-cl({y}). This shows that Mic-cl({x}) 6= Mic-cl({y}). By (4), Mic-cl({x}) ∩ Mic-cl({y}) = φ for each y ∈ U \ K and hence Mic-cl({x}) ∩ ( ⋃ y∈U\K Mic-cl({y})) = φ. On other hand, since K ∈ µR(X) and y ∈ U \K, we have Mic- cl({y}) ⊆ U \K and hence U \K = ⋃ y∈U\K Mic-cl({y}). Therefore, we obtain (U \K) ∩Mic-cl({x}) = φ and Mic-cl({x}) ⊆ K. This shows that U is Micro R0. � Theorem 4.2. Let (U,τR(X),µR(X)) be a micro topological space. Then, U is Micro T1 if and only if U is both Micro T0 and Micro R0. Proof. Necessity. Let L be any Micro open subset of U and x ∈ L. Then by Theorem 3.2, we have Mic- cl({x}) ⊆ L and so by Theorem 3.4, it is clear that U is Micro T0 and Micro R0. Sufficiency. Let x and y be any distinct points of U. Since U is Micro T0, then there exists a Micro open set L such that x ∈ L and y /∈ L. As x ∈ L implies that Mic-cl({x}) ⊆ L. Since y /∈ L, so y /∈ Mic-cl({x}). Hence, y ∈ K = U \Mic-cl({x}) and it is clear that x /∈ K. Thus, it follows that there exist Micro open sets L and K containing x and y respectively, such that y /∈ L and x /∈ K. This implies that U is Micro T1. � Theorem 4.3. Let (U,τR(X),µR(X)) be a micro topological space. Then, the following properties are equivalent: (1) U is Micro R0. (2) x ∈ Mic-cl({y}) if and only if y ∈ Mic-cl({x}), for any points x and y in U. Proof. (1) ⇒ (2): Assume that U is Micro R0. Let x ∈ Mic-cl({y}) and K be any Micro open set such that y ∈ K. Now by hypothesis, x ∈ K. Therefore, every Micro open set which contain y contains x. Hence, y ∈ Mic-cl({x}). (2) ⇒ (1): Let L be a Micro open set and x ∈ L. If y /∈ L, then x /∈ Mic-cl({y}) and hence y /∈ Mic-cl({x}). This implies that Mic-cl({x}) ⊆ L. Hence, U is Micro R0. � Remark 4.1. From Definition 2.10 and Theorem 4.3, the notions of Micro symmetric and Micro R0 are equivalent. Theorem 4.4. Let (U,τR(X),µR(X)) be a micro topological space. Then, U is Micro R0 if and only if for every x and y in U, Mic-cl({x}) 6= Mic-cl({y}) implies Mic-cl({x}) ∩Mic-cl({y}) = φ. Proof. Necessity. Suppose that U is Micro R0 and x,y ∈ U such that Mic-cl({x}) 6= Mic-cl({y}). Then, there exists z ∈ Mic-cl({x}) such that z /∈ Mic-cl({y}) (or z ∈ Mic-cl({y}) such that z /∈ Mic-cl({x})) and there exists K ∈ µR(X) such that y /∈ K and z ∈ K, hence x ∈ K. Therefore, we have x /∈ Mic-cl({y}). Int. J. Anal. Appl. 18 (4) (2020) 581 Thus, x ∈ U \Mic-cl({y}) ∈ µR(X), which implies Mic-cl({x}) ⊆ U \Mic-cl({y}) and Mic-cl({x}) ∩Mic- cl({y}) = φ. Sufficiency. Let K ∈ µR(X) and x ∈ K. We still show that Mic-cl({x}) ⊆ K. Let y /∈ K, that is y ∈ U \K. Then, x 6= y and x /∈ Mic-cl({y}). This shows that Mic-cl({x}) 6= Mic-cl({y}). By assumption, Mic- cl({x}) ∩ Mic-cl({y}) = φ. Hence, y /∈ Mic-cl({x}) and therefore Mic-cl({x}) ⊆ K. Thus, U is Micro R0 � Theorem 4.5. Let (U,τR(X),µR(X)) be a micro topological space. Then, U is Micro R0 if and only if for any points x and y in U, Mker({x}) 6= Mker({y}) implies Mker({x}) ∩Mker({y}) = φ. Proof. Suppose that U is Micro R0. Thus, by Theorem 2.5, for any points x and y in U if Mker({x}) 6= Mker({y}) then Mic-cl({x}) 6= Mic-cl({y}). Now we prove that Mker({x}) ∩ Mker({y}) = φ. Assume that z ∈ Mker({x}) ∩ Mker({y}). By z ∈ Mker({x}) and Theorem 2.3, it follows that x ∈ Mic-cl({z}). Since x ∈ Mic-cl({x}), by Theorem 4.1, Mic-cl({x}) = Mic-cl({z}). Similarly, we have Mic-cl({y}) = Mic- cl({z}) = Mic-cl({x}). This is a contradiction. Therefore, we have Mker({x}) ∩Mker({y}) = φ. Conversely, suppose that for any points x and y in U, Mker({x}) 6= Mker({y}) implies Mker({x}) ∩ Mker({y}) = φ. If Mic-cl({x}) 6= Mic-cl({y}), then by Theorem 2.5, Mker({x}) 6= Mker({y}). Hence, Mker({x})∩Mker({y}) = φ which implies Mic-cl({x})∩Mic-cl({y}) = φ. Because z ∈ Mic-cl({x}) implies that x ∈ Mker({z}) and therefore Mker({x}) ∩ Mker({z}) 6= φ. By hypothesis, we have Mker({x}) = Mker({z}). Then z ∈ Mic-cl({x})∩Mic-cl({y}) implies that Mker({x}) = Mker({z}) = Mker({y}). This is a contradiction. Therefore, Mic-cl({x}) ∩Mic-cl({y}) = φ and by Theorem 4.1, U is Micro R0. � Theorem 4.6. Let (U,τR(X),µR(X)) be a micro topological space. Then, the following properties are equivalent: (1) U is Micro R0. (2) For any non-empty set A and G ∈ µR(X) such that A∩G 6= φ, there exists F ∈ µcR(X) such that A∩F 6= φ and F ⊆ G. (3) For any G ∈ µR(X), we have G = ∪{F ∈ µcR(X): F ⊆ G}. (4) For any F ∈ µcR(X), we have F = ∩{G ∈ µR(X): F ⊆ G}. (5) For every x ∈ U, Mic-cl({x}) ⊆ Mker({x}). Proof. (1) ⇒ (2): Let A be a non-empty subset of U and G ∈ µR(X) such that A ∩ G 6= φ. Then, there exists x ∈ A ∩ G. Since x ∈ G ∈ µR(X), so Mic-cl({x}) ⊆ G. Set F = Mic-cl({x}), then F ∈ µcR(X), F ⊆ G and A∩F 6= φ. (2) ⇒ (3): Let G ∈ µR(X), then G ⊇∪{F ∈ µcR(X): F ⊆ G}. Let x be any point of G. Then, there exists F ∈ µcR(X) such that x ∈ F and F ⊆ G. Therefore, we have x ∈ F ⊆ ∪{F ∈ µ c R(X): F ⊆ G} and hence Int. J. Anal. Appl. 18 (4) (2020) 582 G = ∪{F ∈ µcR(X): F ⊆ G}. (3) ⇒ (4): Obvious. (4) ⇒ (5): Let x be any point of U and y /∈ Mker({x}). Then, there exists K ∈ µR(X) such that x ∈ K and y /∈ K, hence Mic-cl({y}) ∩K = φ. By (4), (∩{G ∈ µR(X): Mic-cl({y}) ⊆ G}) ∩K = φ and there exists G ∈ µR(X) such that x /∈ G and Mic-cl({y}) ⊆ G. Therefore, Mic-cl({x}) ∩G = φ and y /∈ Mic-cl({x}). Consequently, we obtain Mic-cl({x}) ⊆ Mker({x}). (5) ⇒ (1): Let G ∈ µR(X) and x ∈ G. Let y ∈ Mker({x}), then x ∈ Mic-cl({y}) and y ∈ G. This implies that Mker({x}) ⊆ G. Therefore, we obtain x ∈ Mic-cl({x}) ⊆ Mker({x}) ⊆ G. This shows that U is Micro R0. � Corollary 4.1. Let (U,τR(X),µR(X)) be a micro topological space. Then, the following properties are equivalent: (1) U is Micro R0. (2) Mic-cl({x}) = Mker({x}) for all x ∈ U. Proof. (1) ⇒ (2): Suppose that U is Micro R0. By Theorem 4.6, Mic-cl({x}) ⊆ Mker({x}) for each x ∈ U. Let y ∈ Mker({x}), then x ∈ Mic-cl({y}) and by Theorem 4.1, Mic-cl({x}) = Mic-cl({y}). Therefore, y ∈ Mic-cl({x}) and hence Mker({x}) ⊆ Mic-cl({x}). This shows that Mic-cl({x}) = Mker({x}). (2) ⇒ (1): Follows from Theorem 4.6. � Theorem 4.7. Let (U,τR(X),µR(X)) be a micro topological space. Then, the following properties are equivalent: (1) U is Micro R0. (2) If F is Micro closed, then F = Mker(F). (3) If F is Micro closed and x ∈ F , then Mker({x}) ⊆ F . (4) If x ∈ U, then Mker({x}) ⊆ Mic-cl({x}). Proof. (1) ⇒ (2): Let F be Micro closed and x /∈ F. Thus, (U \F) is a Micro open set containing x. Since U is Micro R0, Mic-cl({x}) ⊆ (U \ F). Thus, Mic-cl({x}) ∩ F = φ and by Theorem 2.4, x /∈ Mker(F). Therefore, Mker(F) = F. (2) ⇒ (3): In general, A ⊆ B implies Mker(A) ⊆ Mker(B). Therefore, it follows from (2), that Mker({x}) ⊆ Mker(F) = F. (3) ⇒ (4): Since x ∈ Mic-cl({x}) and Mic-cl({x}) is Micro closed, by (3), Mker({x}) ⊆ Mic-cl({x}). (4) ⇒ (1): We show the implication by using Theorem 4.3. Let x ∈ Mic-cl({y}). Then by Theorem 2.3, y ∈ Mker({x}) and by (4), we obtain y ∈ Mker({x}) ⊆ Mic-cl({x}). Therefore, x ∈ Mic-cl({y}) implies y ∈ Mic-cl({x}). The converse is obvious and U hence is Micro R0. � Int. J. Anal. Appl. 18 (4) (2020) 583 Definition 4.2. Let (U,τR(X),µR(X)) be a micro topological space. Then, U is said to be Micro R1 if for x,y in U with Mic-cl({x}) 6= Mic-cl({y}), there exist disjoint Micro open sets L and K such that Mic-cl({x}) ⊆ L and Mic-cl({y}) ⊆ K. Theorem 4.8. Let (U,τR(X),µR(X)) be a micro topological space. Then, U is Micro R1 if it is Micro T2. Proof. Let x and y be any points of U such that Mic-cl({x}) 6= Mic-cl({y}). By Theorem 3.4 (1), U is Micro T1. Therefore, by Theorem 3.2, Mic-cl({x}) = {x}, Mic-cl({y}) = {y} and hence {x} 6= {y}. Since U is Micro T2, then there exist disjoint Micro open sets L and K such that Mic-cl({x}) = {x} ⊆ L and Mic-cl({y}) = {y}⊆ K. This shows that U is Micro R1. � Theorem 4.9. Let (U,τR(X),µR(X)) be a micro topological space. Then, the following statements are equivalent: (1) U is Micro R1. (2) If x,y ∈ U such that Mic-cl({x}) 6= Mic-cl({y}), then there exist Micro closed sets F1 and F2 such that x ∈ F1, y /∈ F1, y ∈ F2, x /∈ F2 and U = F1 ∪F2. Proof. Obvious. � Theorem 4.10. If U is Micro R1, then U is Micro R0. Proof. Let L be Micro open such that x ∈ L. If y /∈ L, then x /∈ Mic-cl({y}) and Mic-cl({x}) 6= Mic-cl({y}). So, there exists a Micro open set K such that Mic-cl({y}) ⊆ K and x /∈ K, which implies y /∈ Mic-cl({x}). Hence, Mic-cl({x}) ⊆ L. Therefore, U is Micro R0. � Corollary 4.2. Let (U,τR(X),µR(X)) be a micro topological space. Then, U is Micro R1 if and only if for x,y ∈ U, Mker({x}) 6= Mker({y}), there exist disjoint Micro open sets L and K such that Mic-cl({x}) ⊆ L and Mic-cl({y}) ⊆ K. Proof. Follows from Theorem 2.5. � Theorem 4.11. Let (U,τR(X),µR(X)) be a micro topological space. Then, U is Micro R1 if and only if x ∈ U \Mic-cl({y}) implies that x and y have disjoint Micro open neighbourhoods. Proof. Necessity. Let x ∈ U \Mic-cl({y}). Then, Mic-cl({x}) 6= Mic-cl({y}). Thus, x and y have disjoint Micro open neighbourhoods. Sufficiency. First, we show that U is Micro R0. Let L be a Micro open set and x ∈ L. Suppose that y /∈ L. Then, Mic-cl({y})∩L = φ and x /∈ Mic-cl({y}). So, there exist Micro open sets Lx and Ly such that x ∈ Lx, y ∈ Ly and Lx ∩Ly = φ. Hence, Mic-cl({x}) ⊆ Mic-cl(Lx) and Mic-cl({x}) ∩Ly ⊆ Mic-cl(Lx) ∩Ly = φ. Int. J. Anal. Appl. 18 (4) (2020) 584 Therefore, y /∈ Mic-cl({x}). Consequently, Mic-cl({x}) ⊆ L and hence U is Micro R0. Next, we show that U is Micro R1. Suppose that Mic-cl({x}) 6= Mic-cl({y}). Then, we can assume that there exists z ∈ Mic- cl({x}) such that z /∈ Mic-cl({y}). Then, there exist Micro open sets Kz and Ky such that z ∈ Kz, y ∈ Ky and Kz ∩Ky = φ. Since z ∈ Mic-cl({x}), then x ∈ Kz. Since U is Micro R0, we obtain Mic-cl({x}) ⊆ Kz, Mic-cl({y}) ⊆ Ky and Kz ∩Ky = φ. This shows that U is Micro R1. � Conflicts of Interest: The author(s) declare that there are no conflicts of interest regarding the publication of this paper. References [1] S. Chandrasekar, On micro topological spaces, J. New Theory, 26 (2019) 23-31. [2] H. Z. Ibrahim, On new separation axioms via γ-open sets, Int. J. Adv. Res. Technol. 1 (1) (2012), 1-3. 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