On p-open sets in topological spaces

The aim of this paper is to introduce and study some properties of pre--open sets,and study a new class of spaces, called  p -regular space. Determine some properties of  p -regularity and compare with other types of regular spaces. Introduction julian Dontchev, Maximilian Ganster and Takashi Noiri (2000) has introduced the concept of  p -open sets in topological spaces. The purpose of the present paper is to introduced and investige a new separation axiom called p -regular space, by using such sets, we have proved that  p -open and  -open set are identical in  p -regular spaces. Definitions and Preliminaries By a space X we mean a topological space on which no separation axioms is assumed, we recall the following definitions, notational conventions and characterizations. The colsure(resp., Interior)of a subset A of X is denoted by ClA(resp.,IntA). A subset A of X is said to be preopen(Mashour A.S., Abd-El-Monsef M.E. and El-Deeb S.N.,1982) (resp.,pre-regular p-open (Ganster,1987),-open(Velico,1968), p-open (Ganster,2000),-open(Velico ,1968), p-open(Hussein,2003)of a space X, if and only if A InclA(resp., A=pIntApClA,if for each xA,there exist an open(resp.,preopen,open,preopen) set G, such that x G clG A,(resp., x G pclG A, x G IntClG A, x G pIntpClG A).The family of all preopen(resp.,pre-regular p-open ,-open, p-open,-open, p-open) set of a space X,is such that x G clG A,(resp., x G pclG A , x G IntClG A, x G pIntpClGA).The family of all preopen (resp.,pre-regular p-open ,-open, p-open,-open, p-open) set of a space X,is denoted by ) (X PO (resp.PRPO(X),  ) (X O ,  P ) (X O ,O(X), PO(X)). The complemnet of preopen (resp.,pre-regular p-open, -open,p-open,


Introduction
julian Dontchev, Maximilian Ganster and Takashi Noiri (2000) has introduced the concept of  p -open sets in topological spaces.The purpose of the present paper is to introduced and investige a new separation axiom called p  -regular space, by using such sets, we have proved that  p -open and  -open set are identical in  p -regular spaces.

Definitions and Preliminaries
By a space X we mean a topological space on which no separation axioms is assumed, we recall the following definitions, notational conventions and characterizations.The colsure(resp., Interior)of a subset A of X is denoted by ClA(resp.,IntA).A subset A of X is said to be preopen (Mashour A.S., Abd-El-Monsef M.E. and El-Deeb S.N.,1982) (resp.,pre-regularp-open (Ganster,1987),-open (Velico,1968), p-open (Ganster,2000),-open (Velico ,1968), p-open (Hussein,2003)of a space X, if and only if A  InclA(resp., A=pIntApClA,if for each x  A,there exist an open (resp.,preopen,open,preopen) The family of all preopen (resp.,pre-regular p-open ,-open, p-open,-open, p-open) set of a space X,is denoted by ,O(X), PO(X)).The complemnet of preopen (resp.,pre-regular p-open, -open,p-open, -open,p-open)set is called preclosed(resp.,pre-regularp-closed, -closed ,P -closed, -closed,p-closed) set.The intersection of all preclosed (resp.,  -closed, -closed, p-closed)sets containing A is called preclosure (resp.,  -closure and  P -closure, -closure, p-closure).and is denoted by pclA (resp., cl  A and pcl  A, -clA, pcl  A) The union of all preopen (resp.,  -open, -open, p-open)sets contained in A is called preinterior (resp.,  -interior and  P -interior, -interior, p-interior) .and is denoted by pIntA (resp.,  -IntA and pInt  A, -IntA, pInt  A) .A space X is said to be submaximal (Ganster,1987) if and only if every dense subset of X is open set.A space X is said to be pre-T 2 (Mashour et al,1982) if for each x,yX, such that xy,there exist disjoint preopen sets G,H,such that x G and yH.A space X is said to be p * -regular (Ahmad,1990)(resp., p **regular) iff for every xX,and every preclosed set F such that xF,there exist disjoint preopen(resp.,open)set G,H such that x G and FH.

Definition 1:
Let A be any subset of aspce X.A piont xX is in the preclosure of A(briefly x  pclA) (resp., xcl  A, xpcl  A , x pcl  A) if and only if, for each G PO(X)(resp.,G, G PO(X), G PO(X)) containing x, AGφ(resp., AClGφ, ApClGφ, A pcl  Gφ).For properties of definition 1 see (Hussein,2003).

Theorem (1):
The following are equivalent about a space X: 1-X is Alexandroff.

2-Any intersection of open sets is open.
3-Any union of closed sets is closed.

Some properties of pre--open sets
Lemma 1: Each pre--open sets can be written as a union of preopen set.

Proof :
Let A  P O (X) then for each x A there exist BPO (X) s.t.

Remark 1:
The intersection of two  P -open sets need not be  P -open set in general Example 2 :Let X={a, b, c}, τ ={Φ, X, {a, b}}, then {a, c}  P O (X) and {b, c}  P O (X) but {a, c}∩{b, c}={c}  P O (X).Lemma 3: θO (X) and PθO(X) are identical if (X,τ) is submaximal.Proof :From diagram we have every θO(X) is PθO(X),so to show that  O(X) is PθO(X) are identical,we have only to show that PθO(X) is  O(X) .Let A PθO (X) then for all x  A, there exist GPO (X) s.t.xGPclGA, but since X is submaximal, we have PO (X)=τ, so that Gτ therefore PclG=clG, it follows that xGclGA, hence AθO (X).Proof :Let Aτ, we have to show that APθO (X), APO (X) by theorem 3.2.1 [1].For each xA, there exist BPO (X) such that xBPclBA, which implies that APθO (X).
Corollary 2: PO (X) and PθO (X) are identical if X is p * -regular space.
Proposition 2: Let X 1 , X 2 be two topological spaces and X=X 1 x X 2 ,let APθO(X i ) for I=1,2, then A 1 x A 2 Pθ(X 1 x X 2 ).Proof : Let (x 1 , x 2 )A 1 x A 2 then x 1 A 1 and x 2 A 2 since A 1 , A2PθO(X i ) ,there exist preopen sets G 1 ,G 2 such that x 1 G 1 pClG 1 A 1 and .

Proposition 3:
Let (Y,τ Y ) be a subspace of a space (X,τ).If AY and APθO (X) then APO(Y) Proof :Let APθO(X), to show that APO(Y) we have APθO(X) , then for each xA, there exist GPO(X) such that xGPcl x GA,but GPO(X) and GA, but AY, then GY so that GPO(Y) by theorem(3), hence GPcl x G,but G=G∩YPcl x G∩YA∩Y=A so for each xA,there exist GPO(Y)such that xGPcl y GA,so that APθO (Y). .Proof: follows from theorem (4).

Proposition 5:
A space X is pre-T 2 iff for each x,yX, such that xy,there exist preopen sets G,H,such that x G and ypClG.Proof: Obvious.

Theorem (6):
A space X is pre-T 2 if and only if every sengelton set is pre-θ-closed Proof: (Necessity) Let H={a},and let bH,we have ab,since X is pre-T2 by theorem5, there exist a preopen set G such that bG and apClG, pClGH=φ, therefor bpCl  H, it follows that H pre-θ-closed set.(Sufficiency) Let a,bX such that ab,and let H={a},by hypothesis H pre-θ-closed,we have bpCl  H ,there exist a preopen set G such that bG,pClGH=φ,then apClG,aX/ pClG, X/ pClGG=φ,G and X/ pClG are preopen setswhich containing b,a respectively,therefor X is pre-T 2 .

Pθ-regular space Definition 1:
A space X is said to be Pθ-regular iff for each Pθ-closed set F and a point xX such that xF,there exist two open sets G and H such that xG, FHand G∩H=Φ.

Proposition 1:
Each P ** -regular space is Pθ-regular.The converse of the a above lemma is not true in general as the following example show Example 1: Let X={a, b, c}, and τ={Φ, {a},{b},{a, b},X} then PO (X)= τ, and PθO(X)={ Φ, X}, then X is Pθ-regular which is not P ** -regular.

Theorem (1):
For any topological space (X,τ) the following are equivalent: Theorem 4: If X is Pθ-regular space then PθO (X)=θO (X).Proof: Let APθO(X),since X is Pθ-regular,by theorem(4.1ii)foreach xA, There exist an open set G such that xGclGA, therefor AθO (X) The converse part follows from above diagram Theorem 5: A space X is Pθ-regular if PO (X,τ)=τ.Proof: It is not hard and therefore it is omitted.
Theorem 6: Every Pθ-regular and pT o -space (X,τ) is a T 2 -space.Proof: Let x, yX,such that x≠y, since X is preT o -space, then, there exist a preopen set A containing x but not y, A is Pθ-open set containing x but not y, since X is Pθ-regular and xA,  an open set B s.t.xBclBA.
τ) is P * -regular, then every open set is Pθ-open.
For every xX and every Pθ-open set A containing x there exists an open set B such that xBclBA.iii-Every Pθ-closed set F is the intersection of all closed nbd of F. iv-For every non-empty subset A of X and every Pθ-open subset B of X such that A∩B≠Φ, an open sets C of X such that A∩C≠Φ and clCB.v-For every non-empty subset A of X and every Pθ-closed subset F of X such that A∩F=Φ, there exist two open sets B and C s.t.A∩B≠Φ, B∩C=Φ and FC.Proof :(i)(ii) Let A be a Pθ-open set of X containing x, X\A is Pθ-closed subset of X and xX\A by(i) , there exist two open subsets B and C such that xB, X\AC, and B∩C=Φ, therefore xBX\CA hence xBclBcl(X\C)=X\CA which implies that xBclBA (ii)(iii): Let F be Pθ-closed , and xF, then xX\F, and X\F is a Pθ-open subset of X, using(ii) there exists an open set B such that xBclBX\F , hence FX\clBX\B consequently X\B is a closed nbd of F to which x dose not belong , this prove(iii).(iii)(iv) let Φ≠AX ,and B be any Pθ-open subset of X s.t.A∩B≠Φ, let xA∩B, since xX\B is Pθ-closed so there exists a closed nbd of X\B,say E such that xE,let X\BDE,where D is an open set,then C=X\E and xC ,and A∩C≠Φ also X\D being closed,clC=cl (X\ E)X\DB,hence clCB.(iv)(v): let Φ≠AX,and F be any Pθ-closed subset X such that A∩F=Φ, then A∩X\F≠Φ and X\F is Pθ-open subset using (iv) there exists an open subset B of X such that A∩B≠Φ and BclBX\F.Putting C=X\clB then FCX\B and C is open, this implies (v).(v)(i): Let xF where F is Pθ-closed , and let A={x}≠Φ ,then A∩F=Φ, and hence using(v)  two open sets B and C such that A∩B≠Φ, B∩C=Φ, and FC which implies that (X,τ) is Pθ-regular.Proposition 2: A topological space (X,τ) is Pθ-regular iff for every xF, F is Pθ-closed,  two open subsets G and H s.t.xG and FH and clG∩clH=Φ.Proof: The sufficiency follows directly, and necessity follows from theorem4.1ii.Proposition 3: If A is clopen subset of X, then A is Pθ-open set.Proof: If A=Φ, there is nothing to prove,if A≠Φ, let xA, then xApclA=AclIntAA.Theorem 3: If X is Alexandroff and Pθ-regular space, then every pθ-open set A is clopen set.Proof:Let A be Pθ-open set, then by (theorem 4.1 ii ), there exist an open set G x such that xG x clG x A, hence A= {G x , xA }=  A x {clG x , xA } it follows that A is open set since X is Alexandroff space union of any closed set is closed  xA clG x is closed, then A is closed.Therefore A is closed as well as open.