A UNIFIED THEORY FOR WEAK SEPARATION PROPERTIES

We devise a framework which leads to the formulation of a unified theory of normality (regularity), semi-normality (semi-regularity), s-normality (s-regularity), feeblynormality (feebly-regularity), pre-normality (pre-regularity), and others. Certain aspects of theory are given by unified proof.

Throughout the paper, (X, τ) and (Y , ϑ) mean topological spaces on which no separation axioms are assumed unless otherwise explicitly stated.Let A be a subset of X, A is called semi-open [20] (respectively, pre-open [26], feebly open [24], and β-open [1] [22] (respectively, pre-T i , feebly-T i [18], and β-T i [25] [12] if for every semi-open cover of it has a finite subcover.A space (X, τ) is extremally disconnected (E.D.) [5,6] if cl O ∈ τ for each O ∈ τ and is submaximal [5,6] if all dense sets in it are open.A space (X, τ) is sweakly Hausdorff [11] if for each pair x, y ∈ X such that cl{x} ≠ cl{y}, there exist disjoint semi-open sets U and V such that x ∈ U and y ∈ V .A space (X, τ) is R 1 [8] (semi-R 1 [9]) if and only if for each pair x, y ∈ X such that cl{x} ≠ cl{y} (scl{x} ≠ scl{y}), there exist disjoint open (semi-open) sets U and V such that cl{x} ⊆ U and cl{y} ⊆ V (scl{x} ⊆ U and scl{y} ⊆ V ).

A unified framework
Definition 2.1 (see [7]).Let (X, τ) be a topological space.A mapping ϕ : P (X) → P (X) is called an operation on P (X), where P (X) denotes the family all of the subsets of X if and only if for each A ∈ P (X) − {∅}, int A ⊆ A ϕ and ∅ = ∅ ϕ , where A ϕ is denotes the value of ϕ in A. The class of all operations on P (X) is denoted by O(X).
Throughout the paper, all of the operations on P (X) are assumed to be monotonous (i.e., such that A ⊆ B implies A ϕ ⊆ B ϕ ).Definition 2.2 (see [7]).Let (X, τ) be a topological space, G, H ∈ P (X), and ϕ ∈ O(X).
Definition 2.5.A topological space (X, τ) is (i) a ϕ − T 0 -space if, for any two distinct points x and y of X, there exists either a ϕ-open set containing x but not y or a ϕ-open set containing y but not x.
(ii) a ϕ−T 1 -space if, for any two distinct points x and y of X, there exists a ϕ-open set containing x but not y and a ϕ-open set containing y but not x.Definition 2.6.Let (X, τ) be a topological space, and ϕ, ψ ∈ O(X).
Definition 2.9.Let (X, τ) be a topological space and ϕ ∈ O(X).Let R be the equivalence relation on the space (X, τ) defined by xRy if and only if ϕ cl{x} = ϕ cl{y}.The ϕ − T 0 -identification space of (X, τ) is (X ϕ ,Q(X ϕ )), where X ϕ is the set of equivalence classes of R and Q(X ϕ ) is the decomposition topology on X ϕ .Let P ϕ : (X, τ) → (X ϕ ,Q(X ϕ )) denote the natural map.Table 2.1 lists the type of ϕψ-normal, ϕψ-regular, ϕ − R 0 , and ϕ (ψ)-T 0 -identification spaces induced by operations ϕ and ψ, and gives explicitly the weak forms of normality and regularity to which they correspond.Besides, some new definitions appear in the developing of the unified theory.
Proof.Let A be a ϕ-closed set not containing x. Then x ∈ X − A is ϕ-open set, which implies ϕ cl{x} ⊆ X −A, and there exist disjoint ψ-open sets U and V such that x ∈ ϕ cl{x} ⊆ U and A ⊆ V .

open, and
Theorem 3.5.The following are equivalent: Then there exist x, y ∈ X such that A = ϕ cl{x} and B = ϕ cl{y}, and  2.1, Theorem 3.5 represents the unification of various results in the literature.For example, if ϕ = cl • int, we get a result pertaining to semi-R 0 space.Thus, the following are equivalent: [13,Theorem 2.2].The classical result pertaining to R 0 space follows by substituting int for ϕ.Theorem 3.6.A space (X, τ) is ϕψ-normal if and only if its ψ − T 0 -identification space (X ψ ,Q(X ψ )) is ϕψ-normal, where ϕ ≤ ψ.Referring to Table 2.1, Theorem 3.6 contains several results in the literature.For example, with ϕ = ψ = cl • int it gives that a space (X, τ) is semi-normal if and only if its semi-T 0 -identification space (X s ,Q(X s )) is semi-normal, a result due to Dorsett [17].Similarly, with ϕ = int, ψ = cl • int it gives that a space (X, τ) is s-normal if and only if its semi-T 0 -identification space (X s ,Q(X s )) is s-normal, a result due to Dorsett [15].Theorem 3.7.A space (X, τ) is ϕψ-regular if and only if its ψ − T 0 -identification space (X ψ ,Q(X ψ )) is ϕψ-regular, where ϕ ≤ ψ.

Proof. (⇒)
Let A be ϕ-closed set in X ψ and let C x ∉ A, where C x is the equivalence class containing x. Since P ψ is continuous and open, then P −1 ψ (A) is ϕ-closed set in X not containing x. Then there exist disjoint ψ-open sets U and V such that x ∈ U and P −1 ψ (A) ⊆ V .Since P ψ is continuous, open and P −1 ψ (P ψ (U)) = U for all U ∈ ψO(X), then P ψ (U ) and P ψ (V ) are disjoint ψ-open sets in X ψ such that P ψ (x) = C x ∈ P ψ (U) and A ⊆ P ψ (V ).
(⇐) Let A be disjoint ϕ-closed set in X not containing x. Since P ψ is continuous, open and P −1 ψ (P ψ (A)) = A for all A ∈ ψO(X), then P ψ (A) is ϕ-closed set in X ψ and C x ∉ P ψ (A).Then there exist disjoint ψ-open sets U and V in X ψ such that C x ∈ U and P ψ (A) ⊆ V and P −1 ψ (U ) and P −1 ψ (V ) are disjoint ψ-open sets in X containing x and A, respectively.Using Table 2.1, Theorem 3.7 unifies several known results.For example, if ϕ = ψ = cl • int, then it yields Theorem 2.1((a) (b)) of Dorsett [14].Similarly, with ϕ = int, ψ = cl • int it yields Theorem 2.1 of Dorsett [15].
Proof.Let A and B be disjoint ϕ-closed sets in X.Since ϕ ≤ ψ, A and B are disjoint ψ-closed sets and then there exist disjoint ψ-open sets U and V such that A ⊆ U and B ⊆ V .
Referring to Table 2.1, Theorem 3.8 contains several results in the literature.For example, with ϕ = int, ψ = cl • int it gives that if a space (X, τ) is semi-normal, then (X, τ) is s-normal, a result due to Dorsett [17].
Proof.The proof is similar to that of Theorem 3.8.
Referring to Table 2.1, Theorem 3.9 contains several results in the literature.For example, with ϕ = int, ψ = cl • int it gives that if a space (X, τ) is semi-regular, then (X, τ) is s-regular, a result due to Dorsett [14].
Example 3.2 [11] shows that converse of the Theorem 3.9 is false.In [4], it was shown that for a subset A of (X, τ), int A ⊆ α int A ⊆ p int A ⊆ β int A, int A ⊆ α int A ⊆ s int A ⊆ β int A, and in [16], int A ⊆ f int A ⊆ s int A ⊆ β int A, where α int A (respectively, f int A, s int A, p int A, and β int A) is α-interior (respectively, feebly-interior, semi-interior, pre-interior, and β-interior) of A. Later in [3], it was shown that if (X, τ) is submaximal and E.D., then τ = βO(X) ={A ⊆ X | A is β-open set}.Therefore, by Lemma 3.10, if (X, τ) is semicompact R 0 and submaximal, τ = βO(X) = {A ⊆ X | A is β-open set}.Consequently, we obtain the following.
) spaces are defined as ordinary ones except each open set is replaced by semi-open (respectively, pre-open, feebly open, and β-open) one (i) The intersection of all ϕ-closed sets containing G is the ϕ-closure of G, denoted by ϕ cl G. (ii) The union of all ϕ-open subsets of G is the ϕ-interior of G, denoted by ϕ int G.The set ϕ cl G is the smallest ϕ-closed set containing G, and the set ϕ int G is the largest ϕ-open subset of G.In a topological space, if ϕ = int, then ϕ cl = cl.Similarly, if ϕ = int • cl, then ϕ cl = pcl, where pcl denotes pre-closure.
τ) and let x ∈ O. Let y ∉ O and let C x ,C y ∈ X ϕ containing x, y, respectively.Then x ∉ ϕ cl{y}, which implies C x ≠ C y and there exists a ϕopen set A such that C y ∈ A and C x ∉ A. Since P ϕ is continuous and open, then y ∈ B = P −1 ϕ (A) ∈ ϕO(X, τ) and x ∉ B, which implies y ∉ ϕ cl{x}.Thus ϕ cl{x} ⊆ O.According to Table A and B be disjoint ϕ-closed sets in X ψ .Since P ψ is continuous, open and ϕ ≤ ψ, P −1 ψ (A) and P −1 ψ (B) are disjoint ϕ-closed sets in X.Then there exist disjoint ψ-open sets U and V such thatP −1 ψ (A) ⊆ U and P −1 ψ (B) ⊆ V .Since P ψ is continuous, open and P −1 ψ (P ψ (U )) = U for all U ∈ ψO(X), then P ψ (U) and P ψ (V ) are disjoint ψ-open sets in X ψ such that A ⊆ P ψ (U) and B ⊆ P ψ (V ).(⇐)Let A and B be disjoint ϕ-closed sets in X.Since P ψ is continuous, open and P −1 ψ (P ψ (C)) = C for all C ∈ ψO(X), P ψ (A) and P ψ (B) are disjoint ϕ-closed sets in X ψ .Then there exist disjoint ψ-open sets U and V in X ψ such that P ψ (A) ⊆ U and P ψ (B) ⊆ V .Since P ψ is continuous and open, then P −1 ψ (U) and P −1 ψ (V ) are disjoint ψ-open sets in X containing A and B, respectively.

Lemma 3 . 10 .
If (X, τ) is semicompact R 0 , then (X, τ) is E.D.Proof.Let A be an open set in X.Since every open set is also closed [17, Theorem 2.4], A ⊆ int(cl A) and cl A ⊆ int(cl A).Therefore, we obtain cl A is open set in X.