On Minimal and Maximal Regular Open Sets

The purpose of this paper is to investigate 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 if X is a semi-regular space, then miO(X) = miRO(X). We introduce and study new type of sets called minimal regular generalized closed. A special interest type of topological space called rTmin space is studied and obtain some of its basic properties.


Introduction and Preliminaries
A subset A of a topological space (X, τ ) is called a semi-open [14] (resp. a preopen [1], an α-open [16]) set if A ⊆ Cl(Int(A)) (resp. A ⊆ Int(Cl(A)), A ⊆ Int(Cl(Int(A)))). It is called a semi-closed [19] (resp. a preclosed [13], an α-closed [8] ) set if A c is semi-open ( resp. preopen, α-open). The family of all semi-open (resp. preopen, α-open) sets is denoted by SO(X) (resp. P O(X), τ α ). A subset A of a topological space X is said to be a regular open set [10] if A = Int(Cl(A)). It is called a regular closed if A c is a regular open. The family of all regular-open (resp. regular-closed) sets is denoted by RO(X, τ ) or simply RO(X) (resp. RC(X, τ ) or simply RC(X)). In [10], it was shown that the regularly 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 and only if the regularly open sets of (X, τ ) is a base for τ ; that is, τ = τ s . For a space (X, τ ), the regularly open sets of (X, τ ) equal the regularly open sets of (X, τ s ). Hence, the semi-regularization process generates at most one new topology. Thus (τ s ) s = τ s [10].
A point x ∈ X is said to be a δ-cluster point of a subset A 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). A subset A is called δ-closed if A = Cl δ (A). The complement of a δ-closed set is called a δ-open. A space X is said to be a locally finite if each of its elements is contained in a finite open set. And RC(X, τ ) = {Cl(Int(C)) : Theorem 1.2. [9] Let (X, τ ) be a topological space and A a nonempty subspace of X. A proper nonempty open (resp. closed) subset U of X is said to be a minimal open (resp. a minimal closed) set [6] if any open (resp. closed) set which is contained in U is φ or U . A proper nonempty open (resp. closed) subset M of X is said to be a maximal open (resp. a maximal closed) set [7] if any open (resp. closed) set which contains M is X or M . The collection of all minimal open (resp. maximal open, minimal closed, maximal closed) sets is denoted by m i O(X) (resp. M a O(X), m i C(X), M a C(X)).

(a) If W is an open set such that
Theorem 1.5.
[6] Let U be a nonempty proper open set in a topological space X. 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 .
(3) Cl(S) = Cl(U ) for any nonempty subset S of U .  [20] if any regular open set contained in A is A or φ and a minimal regular closed set [11] if any regular closed set contained in A is A or φ.
b) a maximal regular open set [11] if any regular open set contains A is X or A and a maximal regular closed set [20] 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 )). Proof. Let V be a regular closed such that X\F ⊆ V , then X\V ⊆ F . This implies that X\V = φ or X\V = F ; that is, V = X or V = X\F .
Continuing this process only finitely many, we get a minimal regular open set U = V n for some positive integer n.  (1) U is a minimal regular open.     (2) A is dense in X.
Proof. By Remark 1.12, it suffices to prove that if A is not dense, then A is regular open. If A is not dense, then A ⊆ Cl(A) ⊂ X. Then A ⊆ Int(Cl(A)) ⊂ X. As A is maximal open, we get A = Int(Cl(A)).    (2) U ⊆ Cl(S) for any nonempty subset S of U .      (1) If A is mr-g-closed, then there exists a unique minimal regular open set U such that Cl(A) ⊆ U . This implies that A is g-mr closed. That is every mr-g-closed set is also g-mr closed set.

Semi-Regular Spaces and Semi-Regularization
(2) Let A ⊆ X. If there is no minimal regular open set U such that A ⊆ U , then A is g-mr closed, but not mr-gclosed.
Example 4.3. In Example 1.11, {a, b} is g-mr closed, but not mr-g-closed. The set {b} is both mr-g-closed and g-mr closed.  Then, the set {1, 2} is g-min-closed, but not g-mr closed.
Theorem 4.7. Let X be a topological space. If A is a nonempty min-g-closed set, then A is mr-g-closed.
Proof. Since   Example 4.11. In Example 1.11, the set {c, d} is a regular g-closed, but not mr-g-closed.
Theorem 4.12. Let U be a minimal regular open set and A ⊆ U . Then, A is a regular g-closed iff A is an mr-g-closed.
Proof. Assume A = φ is a mr-g-closed, then by Theorem 4.10, A is regular g-closed. Conversely, assume A is regular g-closed. Since A ⊆ U and U is a minimal regular open set, U is regular open such that A ⊆ U . Then Cl(A) ⊆ U .
Theorem 4.13. Let X be a semi-regular space and A a subset of X. Then, A is g-min-closed iff A is g-min-regular closed.
Corollary 4.14. Let (X, τ s ) be the semi-regularization of a topological space (X, τ ), then the following statements are equivalent: (a) A is g-mr closed set in (X, τ ).
(b) A is g-mr closed set in (X, τ s ).
(c) A is g-min-closed set in (X, τ s ).
Proof. Follows directly from Theorem 3.11 and Corollary 3.4.   Proof. If U is closed, by Theorem 2.4, Cl(A) = Cl(U ) = U . Thus, by Theorem 4.10, A is regular g-closed. Conversely, if A is regular g-closed, by Theorem 2.4, Cl(U ) = Cl(A) ⊆ U . Therefore, U is closed.  Theorem 5.2. Let (X, τ ) be a topological space, then the following are equivalent: (1) X is rT min space.   Proof. Let U be a nonempty regular open set in A such that U = A. By Theorem 1.2, there exists a nonempty proper regular open set G in X such that U = G ∩ A. As the space X is rT min , G is a minimal regular open in X. By Theorem 2.9, U will be minimal regular open set in A.