On Minimal and Maximal Regular Open Sets حول المجموعات المفتوحة المنتظمة العظمى و الصغرى

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1 The Islamic University-Gaza اجلامعةاإلسالمي ة-غ زة Research and Postgraduate Affairs Faculty of Science Department of Mathematics شئون البحث العلمي و الدراسات العليا كلية العلوم ماجشتير الرياضيات On Minimal and Maximal Regular Open Sets حول المجموعات المفتوحة المنتظمة العظمى و الصغرى Fadwa M. Nasser Supervised by Dr. Hisham B. Mahdi Associate prof. of Topology A thesis submitted in partial fulfillment of the requirements for the degree of Master of mathematics November/ 2016

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4 Contents Acknowledgments iv Abstract v Introduction 1 1 Preliminaries Basic Topological Concepts Regular Open Sets and Semi-Regularization Nearly Open Sets Types of Minimal and Maximal Sets Minimal and Maximal Open Sets Minimal and Maximal Nearly Open Sets Minimal and Maximal Regular Open Sets Minimal and Maximal Regular Open Sets; Definition and Preliminaries More on Minimal and Maximal Regular Open Sets Semi-Regular Spaces and Semi-Regularization Generalized Minimal Regular Closed Sets rt min space Regular-Minimal and Regular Maximal-Continuous Functions 43 ii

5 4.1 r-min and r-max Continuous Functions; Definitions and Characterizations Relations Between r-min, r-max, and Other Types of Continuous Functions Properties of r-min and r-max Continuous Functions Conclusion 53 Bibliography 54 iii

6 Acknowledgments First, I would like to thank Allah, the most gracious, for giving me power to complete this work. I also need to present special thanks and gratitude to my family, who stood beside me all the way till I reached my goal. Then, I would like to present my great thanks and gratitude to Dr. Hisham B. Mahdi, who answered all of my inquiries and helped me protect the thesis. And I cannot forget the staff at the Mathematics department for their assistance. Finally, I must thank all who helped or encouraged me reach my goal. iv

7 Abstract In this thesis, we study the concepts of minimal and maximal regular open sets and their relations with minimal and maximal open sets. We study several properties of such concepts in a semi-regular space. It is mainly shown that in a semi-regular space, the concepts of minimal open sets and minimal regular open sets are identical. We introduce and study new type of sets called minimal regular generalized closed. A special interest type of topological space called rt min space is studied and obtain some of its basic properties. We introduce new types of continuous functions called regular minimal continuous and regular maximal continuous functions. Moreover, the relation between regular minimal and regular maximal continuous functions and other types of continuous functions are studied and investigated. Finally, we study some properties of these types of continuous functions. v

8 الملخص في ىذه األطروحة تم د ارسة مفاىيم المجموعات المفتوحة المنتظمة الصغرى و العظمى و عالقتيا بالمجموعات المفتوحة الصغرى و العظمى و درسنا العديد من ىذه المفاىيم خصائص في الفضاء شبو المنتظم. تم حيث إثبات انو في الفضاء شبو المنتظم فان المجموعات المفتوحة الصغرى تكافئ المجموعات المفتوحة المنتظمة الصغرى. في ىذه األطروحة تم أيضا إنتاج و د ارسة نوع جديد من المجموعات يسمى المجموعات المغمقة " المعممة المنتظمة الصغرى. تم أيضا د ارسة نوع من الفضاءات التبولوجية يسمى الفضاء و حصمنا عمى بعض خصائصو الرئيسية. " rtmin تم إنتاج أيضا جديدة أنواعا من االقت ارنات المتصمة تسمى االقت ارنات المتصمة الصغرى المنتظمة و االقت ارنات المتصمة العظمى المنتظمة. باإلضافة إلى ذلك تم د ارسة العالقة بين االقت ارنات المتصمة الصغرى و العظمى المنتظمة و األنواع األخرى من االقت ارنات المتصمة. و في الختام تم د ارسة بعض خصائص ىذه األنواع من االقت ارنات المتصمة.

9 Introduction Nakaoka F. and Oda N. [24], [25] introduced and studied the notions of minimal open sets and maximal open sets in topological spaces. As a simulation of these studies, minimal semi-open [17] (resp. minimal α-open [15], minimal θ-open [6]) sets and maximal semi-open [17] (resp. maximal α-open [15], maximal θ-open [6]) sets have been introduced and studied. In 1937, Stone gave a new class of open sets called regular open sets which is used to define the semi-regularization of a topological space. In [16] and [1], the authors define the notion of minimal regular open sets and maximal regular open sets. The aim of this thesis is to expand the study of concepts of minimal and maximal regular open sets and investigate some of their fundamental properties. The thesis consists of four chapters. Chapter one is divided into three sections. In the first section, we give some basic concepts of topological spaces that will be used during this thesis. The second section study the concepts of regular open sets and give some properties that related to the concepts of regular open sets. We study the semi-regularization of a topological space. The third section is devoted to the basic definitions and the results of nearly open sets such as semi-open, preopen, α-open and θ-open. Chapter two is divided into two sections. In the first section, we study the concepts of minimal and maximal open sets and give the definition of generalized minimal closed sets. In addition, we give the definition of T min space and present a class of maps that related to minimal and maximal open sets called minimal continuous and 1

10 maximal continuous functions. In the second section, we deals with the concepts of minimal and maximal semi-open (resp. α-open, θ-open) sets and discuss some of their fundamental properties. The third chapter is divided into five sections. In section one, we study classes of minimal and maximal regular open sets which is based on regular open sets and we percent some of the preliminaries that associated with these types of regular open sets. In section two, we expand the study of the concepts of minimal and maximal regular open sets and give some theorems that related to them. The third section shows that a subset of semi-regular space is minimal regular open if and only if it is minimal open. Thus, all theorems that related to minimal open sets, which studied by Nackaoka F. and Oda N. in 2001, satisfy for minimal regular open in semi-regular space. In section four, new classes of sets called generalized minimal regular closed and minimal regular generalized closed are introduced and investigated. Section five introduce a new class of topological spaces called rt min spaces. In chapter four, we study new classes of continuous functions called regular minimal continuous and regular maximal continuous. Chapter four is divided into three sections. Section one introduce the definitions and some preliminaries of the concepts of regular minimal continuous and regular maximal continuous functions. In the second section, we discuss the relation between regular minimal continuous, regular maximal continuous, and other types of continuous functions. In section three, we introduce more properties and theorems that related to these types of continuous functions. 2

11 Chapter 1 Preliminaries

12 Chapter 1 Preliminaries In this chapter, we give some basic concepts of topological spaces that will be used during this thesis. We study a class of open sets called regular open sets and give some of related properties. Finally, we give definitions and some basic properties of generalized open sets as semi-open, preopen, α-open and θ-open sets. 1.1 Basic Topological Concepts Definition [34] Let X be a nonempty set. A topology τ on X is a collection of subsets of X satisfying the following: (a) X and ϕ belong to τ. (b) Any finite intersection of elements of τ is an element of τ. (c) An arbitrary union of elements of τ is an element of τ. Remark (1) We say (X, τ) (or simply X when no confusion can result about τ) is a topological space. (2) The subsets of X belonging to τ are called open sets. Definition [20]If X is a topological space and E X, we say E is closed if X\E is open. 3

13 Remark A subset A of X may be open, closed, both (clopen) or neither. Definition [34] Let X be a topological space and A X. The closure of A, denoted as A or Cl(A), is the closed set defined as: Cl(A) = {K X : K is closed and A K}. Theorem [34] Let A and B be subsets of X. Then, (1) A Cl(A). (2) If A B, then Cl(A) Cl(B). (3) Cl(ϕ) = ϕ. (4) Cl(Cl(A)) = Cl(A). (5) Cl(A B) = Cl(A) Cl(B). (6) Cl(A B) Cl(A) Cl(B). (7) A is closed iff Cl(A) = A. Definition [34] Let A be a subset of a space X. The interior of A in X, denoted as A or Int(A), is the open set defined as: Int(A) = {G X : G is open and G A}. Theorem [34] Let A and B be subsets of X. Then, (1) Int(A) A. (2) If A B, then Int(A) Int(B). (3) Int(X) = X. (4) Int(Int(A)) = Int(A). (5) Int(A B) = Int(A) Int(B). 4

14 (6) Int(A) Int(B) Int(A B). (7) A is open iff Int(A) = A. Remark The complement of a subset A of a space X, denoted by A c, is the set X\A. Remark [34] In any space X and A X, (Cl(A)) c = Int(A c ). Definition [20] A point x X is an exterior point of A if there exists an open set U containing x such that U X\A. The set of exterior points of A is called the exterior of A and it is denoted by Ext(A). Definition [20] Let X be a topological space, x X and A X. Then x is a cluster (or accumulation, limit) point of A if every open set containing x contains at least one point of A different from x. The set of all cluster points of A is called the derive of A and it is denoted as À. Definition [20] Let X be a space. Then, A X is dense in X if Cl(A) = X. Definition [34] Let x be an element of a space X. A neighborhood of x is a set U which contains an open set V containing x. The collection µ x of all neighborhoods of x is the neighborhood system of x. Remark [34] U is a neighborhood of x iff x Int(U). Definition [34] A neighborhood base of x in a topological space X is a subcollection β x taken from the neighborhood system µ x having the property that each U µ x contains some V β x. The elements of a neighborhood base at x are called basic neighborhoods of x. Theorem [34] Let X be a topological space and suppose a neighborhood base has been fixed at each x X. Then, (a) G X is open iff G contains a basic neighborhood of each of its points. 5

15 (b) F X is closed iff each point x / F has a basic neighborhood disjoint from F. (c) Cl(E) = {x X : each basic neighborhood of x meets E}. (d) Int(E) = {x E : some basic neighborhood of x is contained in E}. Definition [34] Let (X, τ) be a topological space. A base for τ (or for X) is a collection β τ such that τ = { B l B : l β}. That is, β is a base if U τ and x U, B β such that x B U. Theorem [34] A collection β is abase for a topology on X iff (a) X = B β B. (b) Whenever B 1, B 2 β with p B 1 B 2, there is B 3 β with p B 3 B 1 B 2. Definition [34] If (X, τ) is a topological space and A X. Then the collection τ A = {G A : G τ} is a topology for A, called the relative topology for A. The fact that a subset of X is being given this topology is signified by referring to it as a subspace of X. Theorem [34] If A is a subspace of a space X and H A, then : (a) H is open in A iff H = G A, where G is open in X. (b) H is closed in A iff H = K A, where K is closed in X. (c) Cl A (H) = A Cl X (H). (d) Int A (H) A Int X (H). Definition [34] Let X and Y be topological spaces and f : X Y. Then f is continuous at x 0 X if for each neighborhood V of f(x 0 ) in Y, there is a neighborhood U of x 0 in X such that f(u) V. We say f is continuous on X if f is continuous at each x 0 X. 6

16 Theorem [34] If X and Y are topological space and f : X Y, then the following are all equivalent: (a) f is continuous. (b) For each open set H in Y, f 1 (H) is open in X. (c) For each closed set K in Y, f 1 (K) is closed in X. (d) For each E X, f(cl X (E)) Cl Y (f(e)). Definition [34] Let X and Y be topological spaces, and f : X Y be a function from X to Y. Then (a) f is an open function if f(g) is open in Y, whenever G is open in X. (b) f is a closed function if f(h) is closed in Y, whenever H is closed in X. 1.2 Regular Open Sets and Semi-Regularization In 1937, regular open sets were introduced and used to define the semi-regularization of a topological space. In this section, we study the concept of regular open sets in a topological space and study some of its fundamental properties. Moreover, some properties of semi-regularization of a topological space will be discussed in this section. Definition [30] Let A be a subset of a topological space X. Then A is called a regular open set if A = Int(Cl(A)). A set A is called regular closed if A c is regular open; that is, A = Cl(Int(A)). The collection of all regular open (resp. regular closed) sets in a topological space X is denoted by RO(X) (resp. RC(X)). Theorem [34] If A and B are both regular open sets, then A B is regular open set. 7

17 Remark If A and B are both regular open sets in a topological space X, then A B need not be regular open set as shown in the following example: Example Let X = R with standard topology. Let A = (0, 1) and B = (1, 2), then A and B are regular open sets, but A B = (0, 1) (1, 2) is not regular open set. Remark Let X be a space and A X. Then: (1) If A is regular open set, then Cl(A) is regular closed set. (2) If A is regular closed set, then Int(A) is regular open set. Definition The family of all clopen sets of a space X is denoted by CO(X). Theorem [22] CO(X) = RO(X) RC(X). Remark If A is regular open set, then (Cl(A)) c is also regular open set. Theorem Let X be a topological space and A, B subsets of X. Then the following statements are equivalent: (1) A is regular open set. (2) A = Int(Cl(U)) for some open set U [9]. (3) A = Ext(O) for some open set O [10]. (4) A = Int(C) for some closed set C [8]. And all the following statements are equivalent: (1) A is regular closed set. (2) A = Cl(Int(C)) for some closed set C. (3) A = Cl(U) for some regular open set U. (4) A = Cl(O) for some open set O [11]. 8

18 In [30], it was shown that the regular open sets of a space (X, τ) is a base for a topology τ s on X coarser than τ. The space (X, τ s ) was called the semi-regularization space of (X, τ). The space (X, τ) is semi-regular if the regular open sets of (X, τ) is a base for τ; that is, τ = τ s. For a space (X, τ), the regular open sets of (X, τ) equal the regular open sets of (X, τ s ). Hence, the semi-regularization process generates at most one new topology, thus (τ s ) s = τ s. [30] Theorem [22] Let (X, τ s ) be the semi-regularization space of a topological space (X, τ), then CO(X, τ) = CO(X, τ s ). Theorem [4] If A and B are disjoint open sets in (X, τ), then Int(Cl(A)) and Int(Cl(B)) are disjoint open sets in (X, τ s ) containing A and B respectively. Remark For an open set O in (X, τ), Cl τ (O) = Cl τs (O). Theorem [12] Let β be a base for (X, τ), then β s = {Int(Cl(B)) : B β} is a base for τ s. Theorem [22] Let A be a subset of a space X. If A is open or dense in X, then: (a) RO(A, τ A ) = {V A : V RO(X,τ)}. (b) (τ A ) s = (τ s ) A (= {A U : U τ s }). Definition [33] A point x X is said to be a δ-cluster point of the subset A of a space X if A U ϕ for every regular open set U containing x. The set of all δ-cluster points of A is called the δ-closure of A and denoted by Cl δ (A). If A = Cl δ (A), then A is called δ-closed and the complement of a δ-closed set is called δ-open. 1.3 Nearly Open Sets The following definition introduces some classes of near open sets. 9

19 Definition Let X be a topological space. A subset A of X is called: (1) semi-open [18] if A Cl(Int(A)). (2) preopen [23] if A Int(Cl(A)). (3) α-open [26] if A Int(Cl(Int(A))). (4) θ-open [33] if for each x A, there exists an open set G such that x G Cl(G) A. Definition The complement of a semi-open (resp. preopen, α-open, θ-open) set is semi-closed [7] (resp. preclosed [14], α-closed [13], θ-closed [33]) set. The family of of all semi-open (semi-closed, preopen, preclosed, α-open, α-closed, θ-open, θ-closed) sets in a topological space X is denoted by SO(X) (resp. SC(X), P O(X), P C(X), τ α, C(τ α ), O θ (X), C θ (X)). Definition [33] Let A be a subset of a topological space X. The semi-closure (resp. pre-closure, α-closure, θ-closure) of A, denoted by scl(a) (resp. pcl(a), Cl α (A), Cl θ (A)), is the intersection of all semi-closed (resp. preclosed, α-closed, θ-closed) sets containing A. Definition [33] The union of all semi-open (resp. preopen, α-open, θ-open) sets of X contained in A is called the semi-interior (resp. pre-interior, α-interior, θ-interior) of A and is denoted by sint(a) (resp. pint(a), Int α (A), Int θ (A)). Remark Since the intersection of semi-closed sets is semi-closed, we get that scl(a) is the smallest semi-closed set containing A. Furthermore the union of semiopen is semi-open, so sint(a) is the largest semi-open set inside A. Similarly for pcl(a), Cl α (A), Cl θ (A), pint(a), Int α (A) and Int θ (A). Definition A subset A of a space X is called (a) a generalized closed [19] (briefly. g-closed) if Cl(A) U whenever A U and U is open set. 10

20 (b) a regular generalized closed [29] (briefly. regular g-closed) if Cl(A) U whenever A U and U is a regular open set. Definition A topological space X is called: (1) A locally finite if each of its elements is contained in a finite open set. (2) A connected if X cannot be represented as the union of two or more disjoint nonempty open subsets. If X is not connected, then X is disconnected. Definition A function f : X Y is called: (a) an almost continuous [31] if for each x X and for each regular open set V containing f(x), there exists an open set U containing x such that f(u) V. (b) an almost perfectly continuous [32] if f 1 (U) is clopen set in X, for every regular open set U in Y. (c) an almost strongly θ-continuous [27] if for each x X and for each regular open set V containing f(x), there exists an open set U containing x such that f(cl(u)) V. (d) a δ-continuous [28] if for each x X and for each regular open set V containing f(x), there exists a regular open set U containing x such that f(u) V. 11

21 Chapter 2 Types of Minimal and Maximal Sets

22 Chapter 2 Types of Minimal and Maximal Sets In this chapter, we study the concepts of minimal and maximal open sets in topological spaces which are introduced by Nakaoka F. and Oda N. in 2001 and Murkhajee A. [21], obtains some conditions for disconnectedness of a topological space in terms of minimal and maximal open sets. In addition, we study the concepts of generalized minimal closed sets and concepts of T min space. A class of maps called minimal continuous and maximal continuous maps are studied. The notions of minimal and maximal semi-open [17] (resp. α-open [15], θ-open [6]) sets are studied. We study these types of sets and discuss some of its fundamental properties. 2.1 Minimal and Maximal Open Sets Definition Let X be a topological space. A proper nonempty open subset U of X is said to be: (a) a minimal open set [24] if any open set which is contained in U is ϕ or U, and a minimal closed set [24] if any closed set which is contained in U is ϕ or U. (b) a maximal open set [25] if any open set which contains U is X or U, and a 12

23 maximal closed set [25] if any closed set which contains U is X or U. The collection of all minimal open (resp. maximal open, minimal closed, maximal closed) sets in a topological space X is denoted by m i O(X) (resp. M a O(X), m i C(X), M a C(X)). Example Let X = {a, b, c, d} with a topology τ = {ϕ, X, {a}, {a, b}, {c, d}, {a, c, d}}. Then, the set {c, d} is minimal open and the set {a, c, d} is maximal open. Also the set {b, c, d} is maximal closed and the set {b} is minimal closed. Theorem Let X be a topological space and U X. Then, U is minimal open [24] (resp. minimal closed [25]) set if and only if X\U is maximal closed (resp. maximal open) set. Proof. Let U be a minimal open set in X, then X\U is closed set. Let V be a closed set such that X\U V, then X\V U. But X\V is open set contained in the minimal open set U, so X\V = ϕ or X\V = U. This implies that V = X or V = X\U. Therefore X\U is maximal closed set. Similarly, if X\U is maximal closed set, then U is minimal open set. Corollary Let X be a topological space with a, b X. Then we have the following: (1) if {a} is an open set in X, then {a} is a minimal open set and so X\{a} is a maximal closed. (2) if {b} is a closed set, then {b} is a minimal closed set and so X\{b} is a maximal open. Lemma [24] Let (X, τ) be a topological space. (1) If U is a minimal open set and W is an open set such that U W ϕ, then U W. (2) If U and V are minimal open sets such that U V ϕ, then U = V. 13

24 Proof. (1) Let W be an open set such that U W ϕ. Since U is minimal open set and U W is open set with U W U, we have U W = U. Therefore U W. (2) If U V ϕ, then by (1), U V and V U. Therefore U = V. Lemma [25] Let (X, τ) be a topological space. (1) If U is a maximal open set and W is an open set such that U W X, then W U. (2) If U and V are maximal open sets such that U V X, then U = V. Proof. (1) Let W be an open set such that U W X. Since U is maximal open set and U W is open set with U U W, then U W = U. Therefore W U. (2) If U V X, then by (1), U V and V U. Therefore U = V. Theorem [24] Let U be a minimal open set. If x is an element of U, then U W for any open neighborhood W of x. Proof. Let W be an open neighborhood of x such that U W. Then U W is an open set such that U W U (:= U W U and U W U) and U W ϕ. This contradicts our assumption that U is a minimal open set. Theorem [24] Let U be a minimal open set and x U. Then U = {W : W is an open neighborhood of x }. Proof. By Theorem and the fact that U is an open neighborhood of x, we have U {W : W is an open neighborhood of x} U. Therefore, we have the result. Theorem [24] Let U be a nonempty open set. Then the following three conditions are equivalent: (1) U is a minimal open set. (2) U Cl(S) for any nonempty subset S of U. 14

25 (3) Cl(U) = Cl(S) for any nonempty subset S of U. Proof. (1 2). Let S be a nonempty subset of U. By Theorem 2.1.7, for any element x of U and any open neighborhood W of x, we have S = U S W S. Then, we have W S ϕ and hence x is an element of Cl(S). (2 3) For any nonempty subset S of U, we have Cl(S) Cl(U). On other hand, since by part (2) U Cl(S), we have Cl(U) Cl(Cl(S)) = Cl(S). Therefore we have Cl(U) = Cl(S) for any nonempty subset S of U. (3 1) Suppose that U is not minimal open set. Then there exists a nonempty open set V such that V U. Hence there exists an element a U such that a / V. If Cl({a}) V ϕ, then there exists an element z Cl({a}) and z V which is open. This implies that a V which is a contradiction. Thus, Cl({a}) X\V. Since V U Cl(U) and Cl({a}) X\V, then Cl({a}) Cl(U). Theorem [24] Let U be a minimal open set. Then any nonempty subset S of U is a preopen set. Proof. By Theorem 2.1.9, we have Int(U) Int(Cl(S)). Since U is an open set, then we have S U = Int(U) Int(Cl(S)). Thus, S is a preopen set. Theorem [24] Let V be a nonempty finite open set, then there exists at least one (finite) minimal open set U such that U V. Proof. If V is a minimal open set, we set U = V. If V is not minimal open, then there exists a finite open set V 1 such that ϕ V 1 V. If V 1 is a minimal open set, we set U = V 1. If V 1 is not minimal open set, then there exists a finite open set V 2 such that ϕ V 2 V 1 V. Continuing this process, we have a sequence of open sets V V 1 V 2 V k. Since V is a finite set, this process repeats only finitely many. Then finally, we get a minimal open set U = V n for some positive integer n. Corollary [24] Let X be a locally finite space and V a nonempty open set. Then there exists at least one (finite) minimal open set U such that U V. 15

26 Proof. Since V is a nonempty open set, there exists an element x of V and a finite open set V x such that x V x. Since V V x is a finite open set, then by Theorem , we get a minimal open set U such that U V V x V. Theorem [25] Let V be a proper nonempty cofinite open subset of a topological space X. Then, there exists at least one (cofinite) maximal open set U such that V U. Proof. If V is a maximal open set, we set U = V. If V is not maximal open set, then there exists an (cofinite) open set V 1 such that V V 1 X. If V 1 is a maximal open set, we set U = V 1. If V 1 is not maximal open set, then there exists an (cofinite) open set V 2 such that V V 1 V 2 X. Continuing this process, we have a sequence of open sets V V 1 V 2 V k. Since V is a cofinite set, this process repeats only finitely many. Then finally, we get a maximal open set U = V n for some positive integer n. Theorem [25] Let U be a maximal open set and x an element of X\U. Then, X\U W for any open set W containing x. Proof. Since x X\U, we have W U for any open set W containing x. Then, by Lemma 2.1.6, W U = X. Therefore, X\U W. Corollary [25] Let U be a maximal open set. Then, exactly one of the following two statements holds: (1) For each x X\U and each open set W containing x, W = X. (2) There exists an open set W such that X\U W and W X. Proof. If the first statement does not hold, then there exists an element x of X\U and an open set W containing x such that W X. By Theorem , we have X\U W. Corollary [25] Let U be a maximal open set. Then, exactly one of the following two statements holds: 16

27 (1) For each x X\U and each open set W containing x, we have X\U W. (2) There exists an open set W such that X\U = W X. Proof. If the second statement does not hold, then by Theorem , we have X\U W for each x X\U and each open set W containing x. Hence we have X\U W. Theorem [25] Let U be a maximal open set. Then, Cl(U) = X or Cl(U) = U. Proof. Since U is a maximal open set, by Corollary , we have one of the following two cases: Case (1): For each x X\U and each open set W containing x, we have X\U W. Let x be an element of X\U and W an open set containing x. Since X\U W, we have W U ϕ for any open set W containing x. Hence, X\U Cl(U). Since X = U (X\U) U Cl(U) = Cl(U) X, we have Cl(U) = X. Case (2): There exists an open set W such that X\U = W. Since X\U = W is an open set, U is a closed set. Therefore U = Cl(U). Corollary [25] Let U be a maximal open set. Then Int(X\U) = X\U or Int(X\U) = ϕ. Proof. Direct from Theorem Theorem [25] Let U be a maximal open set and S a nonempty subset of X\U. Then Cl(S) = X\U. Proof. Since ϕ S X\U, by Theorem , we have W S ϕ for any element x of X\U and any open neighborhood W of x. Then, X\U Cl(S). Since X\U is a closed set and S X\U, we have that Cl(S) Cl(X\U) = X\U. Therefore, X\U = Cl(S). Theorem [25] Let U be a maximal open set and M a subset of X such that U M. Then, Cl(M) = X. 17

28 Proof. Since U M X, there exists a nonempty subset S of X\Usuch that M = U S. Hence by Theorem , we have Cl(M) = Cl(S U) = Cl(S) Cl(U) X\U U = X. Therefore Cl(M) = X. Theorem [25] Let U be a maximal open set and N a proper subset of X with U N. Then, Int(N) = U. Proof. If N = U, then Int(N) = Int(U) = U. Otherwise N U, so U Int(N). Since U is a maximal open set and Int(N) X is open, we get Int(N) = U. Theorem [25] Let U be a maximal open set and M any subset of X with U M. Then M is preopen set. Proof. If M = U, then M is an open set, and hence a preopen set. Otherwise U M. By Theorem , Int(Cl(M)) = Int(X) = X M. Therefore M is a preopen set. Theorem [21] If X contains a maximal open set G and a minimal open set H such that H G, then X is disconnected. Proof. Since G is maximal open set and H G, we get G H = X. But H is minimal open set and again H G, then we get G H = ϕ. Then the space is disconnected. Corollary [21] If G is a maximal open set and H is a minimal open set of a topological space X with H G, then G and H are clopen sets in X. Theorem [21] If a connected space X has a set G which is both maximal and minimal open, then this set is the only nontrivial open set in the space. Proof. Let H be a nonempty proper open set. Since G G H and G is maximal open, then we have two cases: Case (1) G H = G and so ϕ H G. But G is minimal open, then H = G. Case (2) G H = X. Since X is connected, then G H ϕ and so by Lemma 2.1.5, G H X. Again G is maximal open. This implies that H = G. 18

29 Theorem [21] If A and B are two different maximal open sets of a topological space X where A B is a closed set, then X is disconnected. Proof. Since A and B are different maximal open sets, we have A B = X. We put G = A\(A B) and H = B or G = A and H = B\(A B). We note that G, H are nonempty disjoint open sets with G H = X. So X is disconnected. Theorem [21] If there exists a maximal open set which is not dense in a topological space X, then the space X is disconnected. Proof. Let U be a maximal open set which is not dense in X. By Theorem , U = Cl(U). We write G = U and H = X\Cl(U). So X = G H, G H = ϕ and G, H are open sets.therefore X is disconnected. Definition A subset A of a space X is said to be: (a) a generalized minimal closed [5] (briefly. g- minimal closed) if Cl(A) U whenever A U and U is minimal open set. (b) a minimal generalized closed [3] (briefly. minimal g-closed) if A is contained in a minimal open set U and Cl(A) U. Definition [2] A topological space (X, τ) is said to be T min (resp. T max ) space if every nonempty proper open subset of X is minimal open (resp. maximal open) set. Example Let X = {a, b, c} with the topology τ = {ϕ, X, {a}, {b, c}}. Then (X, τ) is both T min and T max space. Theorem [2] If X is a T min or T max space, then τ = {ϕ, X}, τ = {ϕ, X, A} or τ = {ϕ, X, A, A c } for some nonempty proper subset A of X. Corollary [2] The concepts T min and T max spaces are identical. That is, X is T min if and only if X is T max. Proof. Direct from Theorem

30 Remark We will use the notation T min for both T min and T max. Corollary [2] Let X be a T min space and Y an open subspace of X. Then Y is T min space. Proof. By Theorem , then τ = {ϕ, X}, τ = {ϕ, X, A} or τ = {ϕ, X, A, A c } for some nonempty proper subset A of X. Hence, we have three cases: case (1): τ = {ϕ, X}, then τ Y =X = {ϕ, Y }. Case (2): τ = {ϕ, X, A}, then τ Y =X = {ϕ, Y, A} or τ Y =A = {ϕ, Y }. Case (3): τ = {ϕ, X, A, A c }, then τ Y =X = τ, τ Y =A = {ϕ, Y } or τ Y =A c = {ϕ, Y }. Thus, Y is T min space. Definitions [2] Let X and Y be topological spaces. A map f : X Y is called: (a) a minimal continuous (briefly. a min-continuous) if f 1 (M) is an open set in X for every minimal open set M in Y. (b) a maximal continuous (briefly. a max-continuous) if f 1 (M) is an open set in X for every maximal open set M in Y. Example Let X = Y = {1, 2, 3, 4} with τ X = {ϕ, X, {1}, {4}, {1, 4}, {1, 2}, {1, 3, 4}, {1, 2, 4}} and τ Y = {ϕ, Y, {1}, {2}, {4}, {1, 2}, {1, 4}, {2, 4}, {1, 2, 4}, {1, 2, 3}}. Define f : X Y by f(1) = 4, f(2) = 4, f(3) = 3, f(4) = 1 and g : X Y by g(1) = 2, g(2) = 3, g(3) = 4, g(4) = 1. Then f is min-continuous function and g is max-continuous function. Remark [2] Minimal continuous maps and maximal continuous maps are, in general, independent. Example In Example , f is min-continuous, but not max-continuous because {1, 2, 3} is maximal open in Y, but f 1 ({1, 2, 3}) = {3, 4} which is not open in X. Moreover, g is max-continuous, but not min-continuous because {4} is minimal open set in Y, but g 1 ({4}) = {3} which is not open in X. 20

31 Theorem [2] Let X and Y be topological spaces and A a nonempty subset of X. If f : X Y is a min-continuous (resp. a max-continuous), then the restriction map f A : A Y is min-continuous (resp. max-continuous). Proof. Let M be a minimal open set in Y. Since f : X Y is min-continuous, f 1 (M) is an open set in X and so (f A) 1 (M) = f 1 (M) A is an open set in A. Therefore f A : A Y is min-continuous. Remark [2] The composition of min-continuous (resp. max-continuous ) maps need not be a min-continuous (resp. a max-continuous) map. Example Let (X, τ X ) and (Y, τ Y ) be as in Example Let Z = {1, 2, 3} with a topology τ Z = {ϕ, Z, {1}, {2}, {1, 2}}. Define f : X Y as in Example and h : Y Z by h(1) = 2, h(2) = 2, h(3) = 2 and h(4) = 1. Then f and h are min-continuous maps, but h f : X Z is not min-continuous since {2} is minimal open set in Z, but (h f) 1 ({2}) = {3, 4} which is not open set in X. Define g : X Y as in Example and k : Y Z by k(1) = 3, k(2) = 1, k(3) = 3 and k(4) = 2. Then g and k are max-continuous maps, but k g : X Z is not max-continuous since {1, 2} is maximal open in Z but (k g) 1 ({1, 2}) = {1, 3} which is not open in X. Theorem [2] Let X and Y be topological spaces. If f : X Y is continuous map and g : Y Z is min-continuous (resp. max-continuous), then g f : X Z is min-continuous (resp. max-continuous) Proof. Let U be a minimal open set in Z. Since g : Y Z is min-continuous, then g 1 (U) is open set in Y. Since f : X Y is continuous, then f 1 (g 1 (U)) = (g f) 1 (U) is open in X. Thus, g f : X Z is min-continuous. 2.2 Minimal and Maximal Nearly Open Sets Definition Let (X, τ) be a topological space. 21

32 (1) A proper nonempty semi-open (resp. α-open, θ-open) set U of X is said to be a minimal semi-open[17] (resp. a minimal α-open[15], a minimal θ-open[6]) set if any semi-open (resp. α-open, θ-open) set which contained in U is ϕ or U. A proper nonempty semi-closed (resp. α-closed, θ-closed) set F of X is said to be a minimal semi-closed[17] (resp. a minimal α-closed[15], a minimal θ-closed[6]) set if any semi-closed (resp. α-closed, θ-closed) set which contained in F is ϕ or F. (2) A proper nonempty semi-open (resp. α-open, θ-open) set M of X is said to be a maximal semi-open[17] (resp. a maximal α-open[15], a maximal θ-open[6]) set if any semi-open (resp. α-open, θ-open) set which contains M is X or M. A proper nonempty semi-closed (resp. α-closed, θ-closed) set E of X is said to be a maximal semi-closed[17] (resp. a maximal α-closed[15], a maximal θ- closed[6]) set if any semi-closed (resp. α-closed, θ-closed) set which contains E is X or E. The collection of all minimal semi-open (resp. minimal α-open, minimal θ- open) sets in X is denoted by m i SO(X, τ) (resp. m i τ α (X, τ), m i O θ (X, τ)) and the collection of all maximal semi-open (resp. maximal α-open, maximal θ-open) sets in X is denoted by M a SO(X, τ) (resp. M a τ α (X, τ), M a O θ (X, τ)). Example Let X = {1, 2, 3, 4} with the topology τ = {ϕ, X, {1}, {2}, {4}, {1, 2}, {1, 4}, {2, 4}, {1, 2, 4}, {1, 2, 3}}. Then SO(X, τ) = {ϕ, X, {1}, {2}, {4}, {1, 2}, {1, 4}, {2, 4}, {1, 2, 4}, {1, 2, 3}, {1, 3}, {2, 3}, {2, 3, 4}, {1, 3, 4}}, τ α (X, τ) = {ϕ, X, {1}, {2}, {4}, {1, 2}, {1, 4}, {2, 4}, {1, 2, 4}, {1, 2, 3}} and O θ (X, τ) = {ϕ, X, {4}, {1, 2, 3}}. We have the following: the set {1} is minimal semi-open, the set {1, 3, 4} is maximal semi-open, the set {2} is minimal α-open, the set {1, 2, 4} is maximal α-open, the set {4} is minimal θ-open and the set {1, 2, 3} is a maximal θ-open. Theorem Let U be a proper nonempty subset of X. Then U is a minimal semi-open [17] (resp. a minimal α-open [15], a minimal θ-open [6]) if and only if X\U is a maximal semi-closed (resp. a maximal α-closed, a maximal θ-closed). 22

33 Proof. Let U be a minimal semi-open set in X, then X\U is semi-closed. Let V be a semi-closed set such that X\U V, then X\V U. But X\V is semi-open set contained in a minimal semi-open set U, so X\V = ϕ or X\V = U. This implies that V = X or V = X\U. Therefore X\U is maximal semi-closed set. Similarly, if X\U is maximal semi-closed, then U is minimal semi-open. Remark The collection of minimal semi-open (resp. minimal α-open, minimal θ-open) sets and minimal open sets are, in general, independent. Example In Example 2.2.2, {1, 2, 3} is a minimal θ-open, but not minimal open and {1} is a minimal open, but not minimal θ-open. Remark The collection of maximal semi-open (resp. maximal α-open, maximal θ-open) sets and maximal open sets are, in general, independent. Example Let X 1 = {a, b, c} with a topology τ 1 = {ϕ, X 1, {a}}. SO(X 1, τ 1 ) = {ϕ, X 1, {a}, {a, b}, {a, c}} = τ α (X 1, τ 1 ). Then the set {a} is maximal open, but not maximal semi-open (resp. not maximal α-open) and the set {a, b} is maximal semiopen (resp. a maximal α-open), but not maximal open. Let X 2 = {a, b, c, d} with a topology τ 2 = {ϕ, X 2, {c}, {c, d}, {a, b}, {a, b, c}}. O θ (X 2, τ 2 ) = {ϕ, X 2, {a, b}, {c, d}}. Then the set {a, b} is maximal θ-open, but not maximal open and the set {a, b, c} is maximal open, but not maximal θ-open. Theorem For any topological space X, the following statements hold: (1) Let A be a maximal semi-open (resp. a maximal α-open, a maximal θ-open) set and B a semi-open [17] (resp. an α-open [15], a θ-open [6]) set. If A B X, then B A. (2) Let A and B be two maximal semi-open [17] (resp. maximal α-open [15], maximal θ-open [6]) sets. If A B X, then A = B. Proof. (1) Let A be a maximal semi-open set and B a semi-open set. Assume A B X, then A A B, but A B is a semi-open and A is a maximal semiopen, so A = A B; that is, B A. Similarly if A is a maximal α-open (resp. a 23

34 maximal θ-open) set and B is an α-open (resp. a θ-open) set. If A B X, then B A. (2) Let A and B be a maximal semi-open. If A B X, then from (1) A B and B A, so A = B. Corollary Let A be a minimal semi-closed [17](resp. a minimal α-closed [15], a minimal θ-closed [6]) set in X and x an element of A. Then A B for any semi-open (resp. α-open, θ-open) set B containing x. Proof. Since x A, then B X\A for any θ-open set B containing x. As X\A is maximal θ-open, by Theorem 2.2.8, (X\A) B = X, so A B. Similarly when A is a minimal semi-closed (resp. a minimal α-closed). Theorem Let A be a minimal semi-closed [17] (resp. a minimal α-closed [15], a minimal θ-closed [6]) set. Then, exactly one of the following two statements holds: (1) For each x A, if B is a semi-open (resp. α-open, θ-open) set containing x, then B = X. (2) There exists a semi-open (resp. α-open, θ-open) set B such that A B and B X. Proof. If the first statement does not hold, then there exists an element x of A and a θ-open set B containing x such that B X. By Corollary 2.2.9, we have A B. Theorem Let A be a minimal semi-closed [17] (resp. a minimal α-closed [15], a minimal θ-closed [6]) set. Then, exactly one of the following two statements holds: (1) For each x A, if B is a semi-open (resp. α-open, θ-open) set containing x, then A B. (2) There exists a semi-open (resp. α-open, θ-open) set B such that A = B X. 24

35 Proof. If the second statement does not hold, then, by Corollary 2.2.9, we have A B for each x A and each θ-open set B containing x. Similarly, when A is a minimal semi-closed (resp. a minimal α-closed). Theorem Let U be a maximal semi-open [17] (resp. a maximal α-open [15]) set. Then either scl(u) = U (resp. Cl α (U) = U) or scl(u) = X (resp. Cl α (U) = X). Proof. Let U be a maximal semi-open set. Then by Theorem , we have exactly one of the following two cases: Case (1): For each x X\U, if B is a semi-open set containing x, then X\U B. So, there exists an element z B and z / X\U; that is, z U and so B U ϕ. This implies that x scl(u). Thus, X\U scl(u). Hence, X = U (X\U) U scl(u) = scl(u). Therefore, scl(u) = X. Case (2): There exists a semi-open set B such that X\U = B, so U is semi-closed and so U = scl(u). Corollary Let A be a minimal semi-closed [17] (resp. a minimal α-closed [15]) set. Then either sint(a) = A (resp. Int α (A) = A) or sint(a) = ϕ (resp. Int α (A) = ϕ). Proof. Follows from Theorem and the fact that (scl(a)) c = sint(x\a) and (Cl α (A)) c = Int α (X\A). Theorem Let U be a maximal semi-open [17](resp. a maximal α-open [15]) set and S a nonempty subset of X\U. Then scl(s) = X\U (resp. Cl α (S) = X\U). Proof. Since ϕ S X\U, by Theorem 2.2.9, we have W S ϕ for any element x of X\U and any semi-open set W of x. Then X\U scl(s). Since X\U is semi-closed and S X\U, we have scl(s) scl(x\u) = X\U. Therefore X\U = scl(s). Similarly when U is a maximal α-open set. Theorem Let U be a maximal semi-open [17] (resp. a maximal α-open [15]) set and M X with U M. Then scl(m) = X (resp. Cl α (M) = X). 25

36 Proof. Let U be a maximal semi-open set. Since U M, then there exists a nonempty subset S of X\U such that M = U S. Hence we have scl(m) = scl(u S) = scl(u) scl(s) U (X\U) = X. Therefore scl(m) = X. Similarly when U is a maximal α-open set. Theorem Let U be a maximal semi-open [17](resp. a maximal α-open [15]) set and assume that the subset X\U has at least two elements. Then scl(x\{a}) = X (resp. Cl α (X\{a}) = X) for any element a of X\U. Proof. Follows directly from Theorem Theorem Let U be a maximal semi-open [17](resp. a maximal α-open [15]) set and N be a proper subset of X with U N. Then sint(n) = U (resp. Int α (N) = U). Proof. Let U be a maximal semi-open set. If N = U, then sint(n) = sint(u) = U. Otherwise N U and so U N. It follows that U sint(n). Since U is maximal semi-open, then sint(n) = U. Similarly when U is a maximal α-open set. 26

37 Chapter 3 Minimal and Maximal Regular Open Sets

38 Chapter 3 Minimal and Maximal Regular Open Sets In this chapter, we study and investigate classes of open sets called minimal regular open and maximal regular open sets. We study the relations between these classes and the classes of minimal and maximal open sets. We show that the class of minimal regular open sets and the class of minimal open sets are independent. But in semi-regular spaces, they are identical. We introduce some of the properties of minimal regular open sets in semi-regular spaces. Then, we introduce a new class of sets called generalized minimal regular closed and study some of their fundamental properties. Finally, we study a class of topological spaces called rt min space. 3.1 Minimal and Maximal Regular Open Sets; Definition and Preliminaries In this section, we study a class of minimal and maximal sets based on regular open sets and present some of the preliminaries that associated with this types of open sets. Definitions A nonempty proper regular open set A of a topological space 27

39 (X, τ) is said to be: a) a minimal regular open set [16] if any regular open set contained in A is A or ϕ and a minimal regular closed set [1] if any regular closed set contained in A is A or ϕ. b) a maximal regular open set [1] if any regular open set contains A is X or A and a maximal regular closed set [16] if any regular closed set contains A is X or A. The collection of all minimal regular open (resp. minimal regular closed, maximal regular open, maximal regular closed) sets in a topological space (X, τ) is denoted by m i RO(X, τ) (resp. m i RC(X, τ), M a RO(X, τ), M a RC(X, τ)). Example Let X = {a, b, c, d} with a topology τ = {ϕ, X, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c}, {a, b, d}, {b, c, d}, {b, d}}. Then RO(X, τ) = {ϕ, X, {a}, {c}, {a, c}, {b, d}, {a, b, d}, {b, c, d}}. So, the set {b, d} is minimal regular open and the set {a, c} is maximal regular open. Also the set {c} is minimal regular closed and the set {a, b, d} is maximal regular closed. Theorem Let X be a topological space and F X. Then, F is minimal regular open (resp. minimal regular closed) if and only if X\F is maximal regular closed (resp. maximal regular open). Proof. Let F be a minimal regular open, then X\F is regular closed. Let V be a regular closed such that X\F V, then X\V F. Since X\V is regular open contained in a minimal regular open F, then X\V = ϕ or X\V = F, that is, V = X or V = X\F. Therefore, X\F is maximal regular closed. Conversely, let X\F be a maximal regular closed, then F is regular open. Let U be a regular open such that U F, so X\F X\U. But X\U is a regular closed and X\F is a maximal regular closed, so either X\U = X\F or X\U = X, that is, U = F or U = ϕ. Therefore, F is minimal regular open. Similarly, F is minimal regular closed if and only if X\F is maximal regular open. Remarks Let X be a topological space with a X. Then: 28

40 (1) if {a} is a regular open (resp. regular closed) set, then {a} is minimal regular open (resp. minimal regular closed). (2) if X\{a} is a regular open (resp. regular closed) set, then X\{a} is maximal regular open (resp. maximal regular closed). Remark The collection of all minimal regular open (resp. maximal regular open) sets and minimal open (resp. maximal open) sets are, in general, independent. See the following example: Example In Example 3.1.2, the set {b} is minimal open, but not minimal regular open and the set {b, d} is minimal regular open, but not minimal open. In addition, the set {a, c} is maximal regular open, but not maximal open and the set {a, b, c} is maximal open, but not maximal regular open. Remark If A is a regular open and a minimal open (resp. a maximal open), then A is a minimal regular open (resp. a maximal regular open). Lemma [1] Let X be a topological space. (a) If U is a minimal regular open set and W is a regular open set such that U W ϕ, then U W. (b) If U and V are minimal regular open sets such that U V ϕ, then U = V. Proof. (a) If U W ϕ, then U W U, but U W is a nonempty regular open contained in a minimal regular open set U, so U W = U. Thus, U W. (b) If U V ϕ, then by (a), U V and V U. Thus, U = V. Lemma Let U be a minimal regular open set. If x U, then U W for any regular open set W containing x. Proof. Let x U. If W is a regular open set containing x. Then U W ϕ. So by Lemma part (a), U W. Remark If U is a maximal regular open in X and W is a regular open, then we may have that U W X and W U. Thus if U and V are two different maximal regular open sets, we may have that U V X. See the following example: 29

41 Example Let X = {1, 2, 3, 4, 5} with a base β = {{1}, {2}, {3}, {5}, {3, 4, 5}}. Then, RO(X) = {ϕ, X, {1}, {2}, {3}, {5}, {1, 2}, {1, 3}, {1, 5}, {2, 3}, {2, 5}, {3, 4, 5}, {1, 2, 5}, {1, 2, 3}, {2, 3, 4, 5}, {1, 3, 4, 5}}. Note that {1, 2, 5} and {1, 2, 3} are two maximal regular open sets, but neither {1, 2, 5} {1, 2, 3} = X nor {1, 2, 5} = {1, 2, 3}. Theorem Let V be a nonempty finite regular open set in a topological space X. Then there exists at least one (finite) minimal regular open U such that U V. Proof. If V is a minimal regular open, we set U = V. If V is not minimal regular open, then there exists a (finite) regular open set V 1 such that ϕ V 1 V. If V 1 is a minimal regular open, set U = V 1. If V 1 is not minimal regular open, then there exists a (finite) regular open set V 2 such that ϕ V 2 V 1 V. Continuing this process, we have a sequence of regular open sets V V 1 V 2 V k. Since V is a finite set, then this process repeats only finitely many. So we get a minimal regular open set U = V n for some positive integer n. Corollary Let X be a locally finite semi-regular space. If V is a nonempty open set, then there exists at least one (finite) minimal regular open set U such that U V. Proof. Let V be a nonempty open set with x V. Since X is locally finite space, then there exists a finite open set V x such that x V x, so x V x V which is a finite open. As X semi-regular, there exists a finite regular open set A such that x A V x V. By Theorem , there exists a minimal regular open set U such that U A V x V V. 30

42 3.2 More on Minimal and Maximal Regular Open Sets Theorem If A is a minimal open set in a topological space X such that A is not dense in X, then Int(Cl(A)) is minimal regular open. Proof. Let A be a minimal open set and assume that A is not dense in X. By Theorem 1.2.9, Int(Cl(A)) is regular open. Since A is a nonempty open, then ϕ A Int(Cl(A)). Since Cl(A) X, then Int(Cl(A)) X; that is, Int(Cl(A)) is a nonempty proper regular open set. Let V be a nonempty regular open set such that V Int(Cl(A)). Then, Int(Cl(V )) Int(Cl(A)). If V A = ϕ, then by Theorem , Int(Cl(V )) Int(Cl(A)) = ϕ which is a contradiction. So, V A ϕ. By Lemma part (1), A V and so Int(Cl(A)) Int(Cl(V )). This implies that Int(Cl(V )) = Int(Cl(A)). Hence, V = Int(Cl(A)). Therefore, Int(Cl(A)) is minimal regular open. Remark From Theorem 3.2.1, any minimal open set A such that A is not dense in X, there exists a minimal regular open set U such that A U. Theorem If C is a closed set contained properly in a minimal regular open set U, then Int(C) = ϕ. Proof. Let U be a minimal regular open set and C a closed set such that C U, so Int(C) U. Since Int(C) is regular open and U is minimal regular open, then Int(C) = ϕ. Theorem Let U be a nonempty proper regular open set. Then, the following three conditions are equivalent: (1) U is minimal regular open. (2) U Cl(A), where A U and Int(A) ϕ. (3) Cl(U) = Cl(A), where A U and Int(A) ϕ. 31

43 Proof. (1 2) Let A U such that Int(A) ϕ, then ϕ Int(A) Int(Cl(A)) Int(Cl(U)) = U which is minimal regular open. This implies that U = Int(Cl(A)) Cl(A). (2 3) Since A U, U Cl(A) Cl(U). Therefore, Cl(U) = Cl(A). (3 1) Let B be a nonempty regular open set such that B U, then ϕ Int(B) = B U, so Cl(U) = Cl(B). This implies that B = Int(Cl(B)) = Int(Cl(U)) = U. Therefore, U is minimal regular open. Corollary If U is a minimal regular open and A U such that Int(A) ϕ, then A is preopen. Proof. By Theorem part (2), U Cl(A). So A U = Int(U) Int(Cl(A)). Remark The condition of being Int(A) ϕ is necessary in Theorem and Corollary as shown in the following example: Example In Example 3.1.2, the set {b, d} is minimal regular open, {d} {b, d} and Int({d}) = ϕ, but {b, d} {d} = Cl({d}). In Addition, the set {d} is not preopen. Theorem Let A be a nonempty subspace of X and U a regular open set in A and a regular open in X. If U is a minimal regular open in A, then U is a minimal regular open in X. Proof. U is a nonempty proper regular open set in X. Let S U such that Int X (S) ϕ. Since Int X (S) S U A, so ϕ Int X (S) = Int X (S) A Int A (S). Since U is minimal regular open in A, then by Theorem 3.2.4, U Cl A (S) = Cl X (S) A Cl X (S). Thus, By Theorem 3.2.4, U is minimal regular open in X. Theorem Let A be a nonempty open subspace of X and G a minimal regular open set in X. If G A ϕ and A G, then G A is a minimal regular open in A. 32