ON θ-GENERALIZED CLOSED SETS

The aim of this paper is to study the class of θ-generalized closed sets, which is properly placed between the classes of generalized closed and θ-closed sets. Furthermore, generalized Λ-sets [16] are extended to θ-generalized Λ-sets and R0-, T1/2and T1-spaces are characterized. The relations with other notions directly or indirectly connected with generalized closed sets are investigated. The notion of TGO-connectedness is introduced.

1. Introduction.The first step of generalizing closed sets was done by Levine in 1970 [15].He defined a set A to be generalized closed if its closure belongs to every open superset of A and introduced the notion of T 1/2 -spaces, which is properly placed between T 0 -spaces and T 1 -spaces.Dunham [10] proved that a topological space is T 1/2 if and only if every singleton is open or closed.In [13], Khalimsky, Kopperman, and Meyer proved that the digital line is a typical example of a T 1/2 -space.
Recently, in [8], Ganster and the first author of this paper defined δ-generalized closed sets and introduced the notion of T 3/4 -spaces, which is properly placed between T 1 -spaces and T 1/2 -spaces.They proved that the digital line is T 3/4 .
The aim of this paper is to continue the study of generalized closed sets, this time via the θ-closure operator defined in [26] and characterize T 1/2 -spaces and T 1 -spaces in terms of θ-generalized closed sets.Via θ-closure operator, we extend the class of generalized Λ-sets to the class of θ-generalized Λ-sets and study some new characterizations of R 0 -spaces and T 1 -spaces.

Preliminaries concerning generalized closed sets.
Throughout this paper, we consider spaces on which no separation axioms are assumed unless explicitly stated.The topology of a given space X is denoted by τ and (X, τ) is replaced by X if there is no chance for confusion.For A ⊆ X, the closure and the interior of A in X are denoted by Cl(A) and Int(A), respectively.Sometimes, when there is no chance for confusion, A stands for Cl(A).The θ-interior [26] of a subset A of X is the union of all open sets of X whose closures are contained in A, and is denoted by Int θ (A).The subset A is called θ-open [26] The family of all θ-open sets forms a topology on X and is denoted by τ θ .We use the name CO-set for sets whose closure is open.
) strongly θ-continuous [21] if, for each x ∈ X and each open set V containing f (x), there exists an open set U containing x such that f (U) ⊆ V .

Basic properties of θ-generalized closed sets
We denote the family of all θ-generalized closed subsets of a space (X, τ) by TGC(X, τ).
The next two results together with the examples following them show that the class of θ-generalized closed sets is properly placed between the classes of g-closed and θ-closed sets.
The following diagram is an enlargement of a Diagram from [7].
Proof.Follows easily from Observation 2.1(i) (note that a preopen g-closed set is a CO-set).Lemma 3.9.If A and B are subsets of a topological space (X, τ), then Cl Proposition 3.10.(i) A finite union of θ-g-closed sets is always a θ-g-closed set.(ii) A countable union of θ-g-closed sets need not be a θ-g-closed set.(iii) A finite intersection of θ-g-closed sets may fail to be a θ-g-closed set.
(ii) Let X be the real line with the usual topology.Since X is regular, by Observation 3.5, every singleton in  Example 3.15.Let (X, τ) be the space in the example above.Set H = {a, c, d}.Clearly, H is open in (X, τ) and H is not θ-generalized closed in (X, τ).But B = {a, c} is θ-generalized closed relative to H.However, B is not θ-generalized closed in (X, τ).The notion of a Λ-set and a generalized Λ-set in a topological space was introduced in [16].By definition, a subset A of a topological space (X, τ) is called a Λ-set [16] if

Characterizations of T
Definition 5. (i) For a subset A of (X, τ), we define A Λ θ as follows In [12], A Λ θ is denoted by ker θ A.
Remark 4.5.(i) A Λ-set need not be θ-closed.Any singleton of an infinite space with the cofinite topology is a Λ-set (since the space is T 1 ) but none of the singletons is θ-closed.
(ii) A closed set need not be a Λ-set.In the Sierpinski space (X = {a, b},τ = {∅, {a}, X}), the set B = {b} is closed but B is not a Λ-set.However, in [16,Prop. 3.8], it was shown that in a topological space (X, τ), every subset of X is a generalized Λ-set if and only if every closed set is a Λ-set.
(iii) A generalized Λ-set need not be θ-generalized Λ-set.In an infinite cofinite space X, as mentioned in Remark 4.5, every singleton is a Λ-set and, hence, a generalized Λ-set but none of the singletons is a θ-generalized Λ-set since the θ-closure of every singleton is X.
In [16], it was proved that in T 1 -spaces, every set is a Λ-set.Note that the converse is also true.Proposition 4.6.(i) A topological space (X, τ) is a T 1 -space if and only if every subset of X is a Λ-set.
(ii) A topological space (X, τ) is an R 0 -space if and only if every singleton of X is a generalized Λ-set.

Proof.
(i) Obvious.(ii) In [9], Dube showed that a space is R 0 if and only if, for each closed set A, A = A Λ .Thus, if X is R 0 , then for each singleton {x} and each closed set F containing x, we have {x} ⊆ {x} Λ ⊆ F Λ = F .So, {x} is a generalized Λ-set.For the reverse assume that F ⊆ X is closed.For each x ∈ F , by assumption, {x} Λ ⊆ F .Thus, (ii) In Hausdorff spaces, every subset is a θ-generalized Λ-set.(iii) A topological space X is Hausdorff if and only if X is a kc-space and every closed set of X is a θ-generalized Λ-set.
Example 5.2 and Example 5.4 also show that continuity and θ-g-continuity are independent concepts.Thus, we have the following implications and none of them is reversible.Let g : (X, σ ) → (X, ν) be the identity function.It is easily observed that g is also θgeneralized continuous.But the composition function g • f : (X, τ) → (X, ν) is not θ-generalized continuous since {a, b} ∈ TGC(X, τ).(ii) Let f : (X, τ) → (Y , σ ) be strongly θ-continuous and closed.Then, f is θ-girresolute. Proof.
(i) Left to the reader.(ii) Let B be a θ-g-closed set of (Y , σ ) and let U ∈ τ such that f Ogata [22,Def. 4.12], where γ : τ → ᏼ(X) is the closure operation and id : σ → ᏼ(Y ) is the identity operation.Using [22,Prop. 4.13(i)] and the fact that Cl γ (E) = Cl θ (E) and Cl id (E) = Cl(E) for the closure operation γ, the identity operation id and the subset E, we get (ii) Under the same assumptions of Theorem 5.
Next, we offer the following "Pasting Lemma" for θ-g-continuous functions.

Observation 3 . 1 .Example 3 . 2 .Observation 3 . 3 .
Every θ-closed set is θ-generalized closed.Let X = {a, b, c} and let τ = {∅, {a, b},X}.Set A = {a, c}.Since the only open superset of A is X, A is clearly θ-generalized closed.But it is easy to see that A is not θ-closed.In fact, it is not even semi-closed since its complement {b} has empty interior.Every θ-generalized closed set is g-closed and hence α g-closed, gs-closed, and r-g-closed.Example 3.4.Let X = {a, b, c} and let τ = {∅, {a}, {a, b}, {a, c},X}.Set A = {c}.Clearly, A is closed and hence , b, c, d, e} and let τ = {∅, {a, b}, {c}, {a, b, c},X}.Set A = {a, c, d} and B = {b, c, e}.Clearly, A and B are θ-generalized closed sets since X is their only open superset.But C = {c} = A ∩ B is not θ-generalized closed since C ⊆ {c} ∈ τ and Cl θ (C) = {c, d, e} ⊆ {c}.Proposition 3.11.The intersection of a θ-generalized closed set and a θ-closed set is always θ-generalized closed.Proof.Let A be θ-generalized closed and let F be θ-closed.Let U be an open set such
X\{x} is not open and thus the only superset of B is X.Trivially, B is θ-generalized closed.By (2), B is closed or, equivalently, {x} is open.Thus, every singleton in X is open or closed.Hence, in the notion of [6, Thm.6.2(i)],X is a T 1/2 -space.Let A ⊆ X be θ-generalized closed and let x ∈ Cl θ (A).Since X is T 1 , {x} is closed and thus by Lemma 4.2, x ∈ Cl θ (A)\A.Since x ∈ Cl θ (A), then x ∈ A. This shows that Cl θ (A) ⊆ A or, equivalently, that A is θ-closed.Sufficiency.Let x ∈ X.Assume that {x} is not closed.Then B = X\{x} is not open and, trivially, B is θ-generalized closed since the only open superset of B is X itself.By (2), B is θ-closed and thus {x} is θ-open.Since a singleton is θ-open if and only if it is clopen, {x} is clopen.
1/2 -spaces, T 1 -spaces and R 0 -spaces Theorem 4.1.A space (X, τ) is a T 1/2 -space if and only if every θ-generalized closed set is closed.Proof.Necessity.Let A ⊆ X be θ-generalized closed.By Observation 3.3, A is g-closed.Since X is a T 1/2 -space, A is closed.Sufficiency.Let x ∈ X.If {x} is not closed, then B = Lemma 4.2.Let A ⊆ (X, τ) be θ-generalized closed.Then Cl θ (A)\A does not contain a nonempty closed set.Theorem 4.3.A space (X, τ) is a T 1 -space if and only if every θ-generalized closed set is θ-closed.