On generalized preopen sets

In this paper, the notion of generalized preopen compactness is introduced and connections to other several types of compactness are discussed. In addition, new separation axioms are established


Introduction
Let (X, τ ) be a topological space (or simply, a space) and A ⊆ X.Then A is α-open or α-set [14,12] (resp., semi-open [2], semi-closed [2], preopen [11], preclosed [11], semi-preopen [2], generalized closed [9]) if A ⊆ int(cl(int(A)))(resp., A ⊆ cl(int(A)), int(cl(A)) ⊆ A, A ⊆ int(cl(A)), cl(int(A)) ⊆ A, A ⊆ cl(int(cl(A))), cl(A) ⊆ U , for every open set U containing A), where int( ) and cl( ) are the interior and closure operations, respectively.A is regular-open if A = int(cl(A)) [16].Complements of regular-open sets are called regular-closed.A is semi-regular [11] if it is both semi-open and semi-closed.A is interior closed [7] if int(A) is semiclosed.A is an A-set [16] if A = U ∩ C, where U is an open set and C is a regular closed set.It is known that an A-set is semi-open [16].A is a B-set [4] if A = U ∩ C where U is an open set and C is semi-closed.A is locally where U is open and C is semi-regular.Equivalently, A is an locally closed is open.Spaces that contain two disjoint dense subsets are called resolvable.(X, τ ) is called strongly irresolvable [6] if every open subspace of X is irresolvable, i.e. it can not be represented as a disjoint union of two dense subsets.In [5], it has been pointed out that a space is strongly irresolvable if and only if every preopen set is semi-open.(X, τ ) is said to be P-closed [3] (resp., quasi-H-closed (simply, QHC)) if every preopen (resp., open ) cover of X has a finite subfamily the preclosures (resp., closures) of whose members cover X. (X, τ ) is said to be strongly compact [10] if every preopen cover has a finite subcover.(X, τ ) is called nearly compact [13] if every cover of X by regular open sets has a finite sub cover.Thompson [15] introduced the class of S-closed spaces, where a space X is called S-closed if every semi-open cover of X has a finite subfamily the closures of whose members cover X, or equivalently, if every regular closed cover of X has a finite subcover.(X, τ ) is said to be strongly p-regular [5]

Generalized preopen sets in certain spaces
We begin this section by recalling the concept of generalized preopen set and some related results from [1].

Definition 1. [1]
A subset A of a space X is generalized preopen (simply, gpo-) set if cl(A) ⊆ U , whenever U is a preclosed subset such that U ⊇ A.

Lemma 1. [1]
A subset A of a space X is gpc if and only if for every preopen set V contained in A, V ⊆ int(A).
On generalized preopen sets

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We remark that if A and B are gpo-sets, then A ∩ B need not be a gpo-set, while arbitrary unions of gpo-sets are gpo-sets.
Example 1.Since (1,3], as a subspace of the reals with the standard topology, is a gpo-set but not preopen, and (0,1] with the indiscrete topology is preopen but not a gpo-set.
Then X is an ED space, {c} is a preclosed set that is not a gpo-set.
Moreover, {c} is not a regular closed set.
Definition 2. A space (X, τ ) is gpo-irresolvable if every preopen set is a gpo-set.
Theorem 1.In a strongly irresolvable space, every semi-open set is a gposet.
Proof: Let A be a preopen set.Then since X is strongly irresolvable, A is semi-open and by Theorem 2, A is a gpo-set.
The converse of Theorem 2 is not true in general.A = (−∞, 1] with the leftray topology on the reals is a gpo-set and a preopen set but not semi-open.
The following is a new characterization of open sets.

Theorem 3. [1] A subset A ⊆ X is open if and only if
A is a gpo-set and a preopen set.

Gpo-compact spaces
Several types of compact spaces were discussed in [3,4,5,8,11].In this section, gpo-compact notion is introduced and connections to other several well-known types of compactness are discussed.
Definition 3. A space (X, τ ) is gpo-compact if every gpo-cover (a cover consisting of gpo-sets) of X has a finite subcover.
Equivalently, (X, τ ) is gpo-compact if every gpo-cover of X has a finite subcover.A submaximal space is an example of a gpo-compact space.The proof of the following result follows from the fact that every open set is a gpo-set.
Theorem 4. If (X, τ ) is a gpo-compact space, then it is compact.
Since every compact space is nearly compact and a QHC-space, a gpocompact space is nearly compact and QHC.
Theorem 5.If a space (X, τ ) is gpo-irresolvable and gpo-compact, then it is strongly compact.
Proof: Let A = {A α : α ∈ ∆} be a preopen-cover of X.Then since X is gpo-irresolvable, by Theorem 1, A is a gpo-cover of X and since X is gpo-compact, it has a finite subcover.Thus X is strongly compact.
The converse of Theorem 4 need not be true since a gpo-set need not be preopen.In addition, the notions of gpo-irresolvable and gpo-compact are independent.
Corollary 1.If a space (X, τ ) is gpo-irresolvable and gpo-compact, then it is P-closed.
Proof: Let A = {A α : α ∈ ∆} be a preopen cover of X.Since X is gpo-irresolvable, A α is a gpo-set for all α ∈ ∆ and A is a gpo-cover of X.
Since X is gpo-compact, it has a finite subcover.Thus , and so X is P-closed.
Since a gpo-set need not be preopen, a P-closed space need not be gpo-irresolvable or gpo-compact.
Corollary 2. If (X, τ ) is P-closed, T 0 and gpo-compact, then it is strongly compact.
Proof: Let A = {A α : α ∈ ∆} be a semi-open cover of X.Then A is a gpo-cover of X.Since X is gpo-compact, it has a finite subcover such that Since an s-closed space is S-closed, a gpo-compact space is S-closed.
Thus it has a finite subcover.That is Definition 4. A space (X, τ ) is strongly s-compact if for every gpo-cover cl(A αi ) for every A αi ⊆ X, i = 1, 2..., n.
Definition 5. A space (X, τ ) is strongly Ogpo-regular if X has a gpocover A = {A α : α ∈ ∆} for all x ∈ X, and for all A α (x) ∈ A such that x ∈ A α (x), there exists U x ∈ gpo(X) such that x ∈ U x ⊆ cl(U x ) ⊆ A α (x).
Proof: Let A = {A α : α ∈ ∆} be a gpo-cover of X.Then since X is strongly Ogpo-regular, for all x ∈ X, there exists A α (x) ∈ A such that x ∈ A α (x) and there exists U x ∈ gpo(X) with x ∈ U x ⊆ cl(U x ) ⊆ A α (x).Thus {U x : x ∈ X} is a gpo-cover of X.Since X is strongly gpo-compact, there where U is open and C is closed.Clearly every A-set is a locally closed set and every locally closed set is a B-set.Since regular closed sets are semi-regular and since semi-regular sets are semi-closed, the following implications are obvious: A − set ⇒ AB − set ⇒ B − set, but none of them of course is reversible [4].Moreover, since the intersection of an open set and a semi-regular set is always semi-open, every AB-set is semi-open.A space (X, τ ) is called a partition space if every open subset of X is closed, see [4].(X, τ ) is called hyper connected if every open subset of X is dense.(X, τ ) is called extremely disconnected (simply, ED) if every open subset of X has an open closure or equivalently if every regular closed set (resp., p-regular, almost p-regular) if for each point x ∈ X and each pre-closed set (resp., closed set, regular closed set) F such that x / ∈ F , there exist disjoint preopen sets V and U such that x ∈ U and F ⊆ V .(X, τ ) is called s-closed, if every semi-open cover has a finite subfamily the semi-closures of whose members cover X.Throughout this paper, (X, =) and (Y, δ) stand for topological spaces with no separation axioms assumed unless otherwise mentioned.The fundamental notion of generalized open sets was introduced and explored by several authors.In recent years a number of other generalizations of open sets have been studied.Generalized open sets play a very important role in General Topology and they are now the research topics of many topologists worldwide.In fact, a significant theme in General Topology and Real analysis concerns the variously modified forms of continuity, separation axioms etc. by utiliaing generalized open sets.One of Talal Al-Hawary the most well known notions and also an inspiration source is the notion of generalized preopen sets introduced in [1].This class is a superset of the class of semi-closed sets, the class of α−sets, the class of AB-sets, the class of A-sets and the class of semi-regular sets.Moreover, these investigations lead to solve the problem of finding the continuity dual of some generalized continuous functions in order to have a decomposition of continuity.This paper is divided as follows: In Section 2, the concept of generalized preopen set is introduced and explored.Properties and connections to other well-known weak and generalized open sets are discussed.Moreover, a characterization of closed and open sets is provided and spaces are studied.In Section 3, generalized preopen compactness is introduced and connections to other types of compactness are discussed.Finally, Section 4 is devoted to discussing new separation axioms defined via generalized preopen sets.
exist U x1 , U x2 , ..., U xn such that X ⊆ Corollary 3. If (X, τ ) is strongly Ogpo-regular, then it is strongly gpocompact if and only if it is gpo-compact.