P-NORMAL,ALMOST P-NORMAL AND MILDLYP-NORMAL SPACES

Govindappa NAVALAGI

Department of MathematicsKLE Society’s

G.H.College, Haveri-581 110Karnataka, INDIA

e-mail : gnavalagi@hotmail.com

Abstract. In 1982 , A.S.Mashhour et al have defined and study the pre-open sets and pre- continuity in topology.Since then many topologists havebeen studied the signi- ficances of preopen sets in the literature.In thispaper, by using preopen sets, we define new separation axioms called al-most p-normality and mild p-normality and also, we continue the studyof p-normality due to [21-22].We show that these three axioms are regularclosed hereditary. We define also the class of almost preirresolute func-tions and show that p-normality is invariant under almost preirresolute M-preopen continuous surjections. Among other things, we prove that almostregular strongly compact space is almost p-normal and weakly p-regularp1-paracompact space is mildly p-normal.

1. INTRODUCTION

Preopen sets sets and precontinuous functions were introduced in 1982 byA.S. Mashhour et al [13].Since then these concepts have been used to defineand to investigate many new topological properties.The purpose of this paperis to examine the normality by defining the concepts of almost p-normality andmild p-normality in terms of preopen sets.In other words, present paper is thestudy of generalization of normality, almost normality and mildly normalityconcepts via preopen sets.

1991 Mathematics Subject Classification. 54A05 ,54B05 , 54C08 , 54D10 ,54D18.Key words and phrases. Preopen sets , semiopen sets,semipreopen sets, α-open, almost

normal, mildly normal, M-preclosed, M-preopen, rc-continuous,preirresolute, α-closed.1

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2. PRELIMINARIES

Throughout the present paper, (X, τ) and (Y, σ) ( or simply X and Y )alwaysmean topological spaces or spaces on which no separation axioms are assumedunless explicitly stated and f : X → Y denotes a single valued function of aspace X into a space Y .Let A be a subset of a space X. The closure and theinterior of A are denoted by Cl(A) and Int(A) , respectively . A subset of aspace X is called regular open(resp. regular closed) if A = Int(Cl(A))( resp.A = Cl(Int(A))).

DEFINITION 2.1:A subset A of a space X is said to be :(i) an α-open [19] if A ⊂ IntClInt(A) , (ii)a semiopen[ 11 ] if A ⊂ ClInt(A)

, (iii)a preopen [13 ] if A ⊂ IntCl(A) ,The family of all α-open ( resp. semiopen , preopen ) sets of a space X

is denoted by αO(X) ( resp. SO(X) ,PO(X)). It is clear that , αO(X) ⊂SO(X) and αO(X) ⊂ PO(X) and moreover, every open set is α-open set(resp. semiopen set , preopen set ) but not conversely.

The complement of a semiopen (resp.preopen , α-open) set is called semi-closed [2]( resp. preclosed [5] , α-closed [25])

It is known that the space (X, τ) is called an α-space [10]if τ = αO(X) andthat a space X is called submaximal [8]if every dense subset of X is open,equivalently, a space X is called submaximal if every preopen set of X is open,since every dense set of a space X is preopen in X.

DEFINITION 2.2 : The intersection of all semiclosed (resp. preclosed) sets containing a subset A of space X is called the semiclosure [2]( resp.preclosure [ 5]) of A and is denoted by sCl(A) ( resp. pCl(A) ).

DEFINITION 2.3[ 15,17]: The union of all preopen sets which are con-tained in A is called the preinterior of A and is denoted by pInt(A) .

DEFINITION 2.4[17,23]: Let X be a topological space and A ⊂ X .Then A is said to be a pre-nieighbourhood of a point x ∈ X, if there exists apreopen set G in X such that x ∈ G ⊂ A.

DEFINITION 2.5[12]:A space is called pre-T2 if for any two distinctpoints x,y of X, there exist disjoint preopen sets U and V such that x ∈ Uand y ∈ V .

DEFINITION 2.6[31]:A space X is said to be weakly Hausdorff if eachpoint of X is the intersection of regular closed sets of X.

DEFINITION 2.7[3]:A space X is called Ro-space iff for each open set Gand x ∈ G , Cl{x} ⊂ G.

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DEFINITION 2.8[5]: A space X is said to be p-regular if for each closedset F of X and each point x ∈ X \F , there exist disjoint preopen sets U andV such that x ∈ U and F ⊂ V .

DEFINITION 2.9[27]:A space X is said to be almost regular if for eachregular closed set F of X and each point x ∈ X \F , there exist disjoint opensets U and V such that x ∈ U and F ⊂ V .

DEFINITION 2.10[12]: A space X is said to be almost p-regular if foreach regular closed set F of X and each point x ∈ X \ F , there exist disjointpreopen sets U and V such that x ∈ U and F ⊂ V .

DEFINITION 2.11[27 ]:A space X is said to be weakly regular if for eachpair consisting of a regular closed set A and a point x such that A ∩ {x} = ∅,there exist disjoint open sets U and V such that x ∈ U and A ⊂ V .

DEFINITION 2.12[28]:A space X is called almost normal(resp. mildlynormal) if for every pair of disjoint closed (resp.regular closed) subsets F andH of X , there exist disjoint open sets U and V such that F ⊂ U and H ⊂ V .

DEFINITION 2.13[16 ]:A space X is called strongly compact if everycover of X by preopen sets has a finite subcover.

DEFINITION 2.14[32]: A subset A of a space X is S-closed relative to Xif every cover of A by semiopen sets of X has a finite subfamily whose closurescover A.

DEFINITION 2.15: A space X is called paracompact [6]( resp.nearlyparacompact[26] , p1 - paracompact [7,15])iff every open (resp. regular open,preopen) covering of X has an open locally finite refinement.

DEFINITION 2.16: A function f : X → Y is said to be :(a) pre-irresolute [24]if inverse image of each preopen set of Y is preopen in

X ,(b) rc-conitnuous [10] if inverse image of each regular closed set of Y is

regular closed set in X.(c) M-preopen [15]if the image of each preopen set of X is preopen in Y ,(d) M-preclosed[15] if the image of each preclosed set of X is preclosed in

Y .(e) α-closed [25] if the image of each closed set of X is α-closed in Y .

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3. P-NORMAL SPACES

We recall the the following.

DEFINITION 3.1[21-22]:A space X is said to be p-normal if for any pairof disjoint closed sets F1 and F2,there exist disjoint preopen sets U and V suchthat F1 ⊂ U and F2 ⊂ V .

Clearly, every normal space is p-normal as every open set is preopen, butnot conversely.For,

EXAMPLE 3.2:Consider the topology τ = {∅, {b, d}, {a, b, d}, {b, c, d}, X}on the set X = {a, b, c, d}. We observe that {∅, {a}, {c}, {a, c}, X} is the familyof closed sets and PO(X) = {∅, {b}, {d}, {a, b}, {a, d}, {b, c}, {b, d}, {c, d},{a, b, c}, {a, b, d}, {a, c, d}, {b, c, d}, X}.

Then X is p-normal, but it is not normal since the pair of closed sets {a}and {c} have no disjoint open-neighbourhoods.

We have the following characterizations of p-normality.

THEOREM 3.3 : For a space X the following are equivalent:(a) X is p-normal.(b)For every pair of open sets U and V whose union is X, there exist pre-

closed sets A and B such that A ⊂ U , B ⊂ V and A ∪B = X.(c)For every closed set F and every open set G containing F , there exists a

preopen set U such that F ⊂ U ⊂ pCl(U) ⊂ G.

PROOF .a⇒b: Let U and V be a pair of open sets in a p-normal spaceX such that X = U ∪ V . Then X \ U , X \ V are disjoint closed sets.Since Xis p-normal there exist disjoint preopen sets U1 and V1 such that X \ U ⊂ U1

and X \ V ⊂ V1. Let A = X \ U1, B = X \ V1. Then A and B are preclosedsets such that A ⊂ U , B ⊂ V and A ∪B = X.

b⇒c: Let F be a closed set and G be an open set containing F . ThenX \F and G are open sets whose union is X.Then by (b), there exist preclosedsets W1 and W2 such that W1 ⊂ X \ F and W2 ⊂ G and W1 ∪ W2 = X.Then F ⊂ X \ W1, X \ G ⊂ X \ W2 and (X \ W1) ∩ (X \ W2) = ∅. LetU = X \W1 and V = X \W2. Then U and V are disjoint preopen sets suchthat F ⊂ U ⊂ X \V ⊂ G. As X \V is preclosed set, we have pCl(U) ⊂ X \Vand F ⊂ U ⊂ pCl(U) ⊂ G.

(c)⇒(a): Let F1 and F2 be any two disjoint closed sets of X. Put G = X\F2,then F1 ∩ G = ∅. F1 ⊂ G where G is an open set. Then by (c), there existsa preopen set U of X such that F1 ⊂ U ⊂ pCl(U) ⊂ G. It follows thatF2 ⊂ X \ pCl(U) = V , say, then V is preopen and U ∩ V = ∅. Hence F1 andF2 are separated by preopen sets U and V . Therefore X is p-normal.

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THEOREM 3.4 : A regular closed subspace of a p-normal space is p-normal.

PROOF . Let Y be a regular closed subspace of a p-normal space X. LetA and B be disjoint closed subsets of Y . As Y is regular closed and henceclosed, A, B are closed sets of X. By p-normality of X, there exist disjointpreopen sets U and V in X such that A ⊂ U and B ⊂ V . As every regularclosed set is semiopen, by Lemma 6.3 in [5], U ∩ Y and V ∩ Y are preopen inY such that A ⊂ U ∩ Y and B ⊂ V ∩ Y . Hence Y is p-normal.

A p-normal space need not be p-regular [5], as the following example shows.For,

EXAMPLE 3.5 :Let X = {a, b, c} and τ = {∅, {a}, {b}, {a, b}, X}. ThenX is a p-normal space, but it is not p-regular since for the closed set {b, c} andthe point a 6∈ {b, c}, there do not exist disjoint preopen sets containing them.

However we observe that every p-normal R0 [3] space is p-regular [5].

The concept of pre-neighbourhood of a subset of a space is given in [17].

Now, we define the following.

DEFINITION 3.6 : A function f : X → Y is said to be almost-preirresoluteif for each x in X and each pre-neighbourhood V of f(x), pCl(f−1(V )) is apre-neighbourhood of x.

Clearly every preirresolute function [10] is almost preirresolute.

The proof of the following lemma is straightforward and hence omitted.

LEMMA 3.7 : For a function f : X → Y , the following are equivalent:(a)f is almost preirresolute.(b) f−1(V ) ⊂ pCl(pCl(f−1(V )))) for every V ∈ PO(Y ).

Now we prove the following.

THEOREM 3.8 :A function f : X → Y is almost preirresolute if and onlyif f(pCl(U)) = pCl(f(U)) for every U ∈ PO(X).

PROOF .Let U ∈ PO(X).Suppose y 6∈ pCl(f(U)).Then there exists V ∈PO(y) such that V ∩ f(U) = ∅.Hence f−1(V )∩U = ∅. Since U ∈ PO(X), wehave pCl(pCl(f−1(V )))∩pCl(U) = ∅.Then by lemma 3.7, f−1(V )∩pCl(U) = ∅and hence V ∩ f(pClU)) = ∅.This implies that y 6∈ f(pCl(U)).

Conversely, if V ∈ PO(Y ), then W = X \ pCl(f−1(V )) ∈ PO(X). Byhypothesis, f(pCl(W )) ⊂ pCl(f(W )) and hence X \ pC(pCl(f−1(V ))) =pCl(W ) ⊂ f−1(pCl(f(W ))) ⊂ f(pCl[f(X \ f−1(V ))]) ⊂ f−1[pCl(Y \ V )] =

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f−1(Y \V ) = X \f−1(V ). Therefore , f−1(V ) ⊂ pCl(pCl(f−1(V ))). By lemma3.7 , f is almost preirresolute.

Now we prove the following result on the invariance of p-normality.

THEOREM 3.9 : If f : X → Y is an M-preopen continuous almostpreirresolute function from a p-normal space X onto a space Y , then Y isp-normal.

PROOF . Let A be a closed subset of Y and B be an open set containingA. Then by continuity of f , f−1(A) is closed and f−1(B) is an open set ofX such that f−1(A) ⊂ f−1(B). As X is p-normal, there exists a preopen setU in X such that f−1(A) ⊂ U ⊂ pCl(U) ⊂ f−1(B) by Theorem 3.3. Then,f(f−1(A)) ⊂ f(U) ⊂ f(pCl(U)) ⊂ f(f−1(B)).Since f is M-preopen almostpreirresolute surjection, we obtain A ⊂ f(U) ⊂ pCl(f(U)) ⊂ B. Then againby Theorem 3.3 the space Y is p-normal.

The following lemma may be proved by using arguments similar to thosegiven in [6,1.4.12,p.52].

LEMMA 3.10 :A function f : X → Y is M-preclosed if and only if foreach subset B in Y and for each preopen set U in X containing f−1(B), thereexists a preopen set V containing B such that f−1(V ) ⊂ U .

Now we prove the following.

THEOREM 3.11 : If f : X → Y is an M-preclosed continuous functionfrom a p-normal space onto a space Y , then Y is p-normal.

Proof of the theorem is routine and hence omitted.

Now in view of lemma 2.2 [9] and lemma 3.10, we prove that the followingresult.

THEOREM 3.12: If f : X → Y is an M-preclosed map from a weaklyHausdorff p-normal space X onto a space Y such that f−1(y) is the S-closedrelative to X for each y ∈ Y , then Y is pre-T2.

PROOF . Let y1 and y2 be any two distinct points of Y . Since X is weaklyHausdorff, f−1(y1) and f−1(y2) are disjoint closed subsets of X by lemma2.2[9].As X is p-normal, there exist disjoint preopen sets V1 and V2 such thatf(yi) ⊂ Vi, for i = 1, 2. Since f is M-preclosed, there exist preopen sets U1 andU2 containing y1 and y2 such that f−1(Ui) ⊂ Vi for i = 1, 2. Then it followsthat U1 ∩ U2 = ∅. Hence the space is pre-T2.

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Next we prove the following.

THEOREM 3.13 : If f : X → Y is an α-closed continuous surjection andX is normal, then Y is p-normal.

PROOF . Let A and B be disjoint closed sets of Y . Then f−1(A) andf−1(B) are disjoint closed sets of X by continuity of f . As X is normal, thereexist disjoint open sets U and V in X such that f−1(A) ⊂ U and f−1(B) ⊂ B.By Proposition 6 in [25], there are disjoint α-open sets G and H in Y such thatA ⊂ G and B ⊂ H. Since every α-open set is preopen, G and H are disjointpreopen sets containing A and B respectively. Therefore, Y is p-normal.

4. ALMOST P-NORMAL SPACES

DEFINITION 4.1 :A space X is said to be almost p-normal if for eachclosed set A and each regular closed set B such that A ∩ B = ∅, there existdisjoint preopen sets U and V such that A ⊂ U and B ⊂ V .

Clearly, every p-normal space is almost p-normal, but the converse is nottrue in general.For,

EXAMPLE 4.2 : Consider the topology τ = {∅, {a}, {a, b}, {a, c}, X} onthe set X = {a, b, c}. Then X is almost p-normal space, but it is not p-normalsince the pair of disjoint closed sets {b} and {c} have no disjoint preopen setscontaining them.

Now, we have characterization of almost p-normality in the following.

THEOREM 4.3 :For a space X the following are statement are equivalent:(a)X is almost p-normal(b)For every pair of sets U and V , one of which is open and the other is

regular open whose union is X, there exist preclosed sets G and H such thatG ⊂ U , H ⊂ V and G ∪H = X.

(c)For every closed set A and every regular open set B containing A, thereis a preopen set V such that A ⊂ V ⊂ pCl(V ) ⊂ B.

PROOF .(a)⇒(b): Let U be an open set and V be a regular open set in analmost p-normal space X such that U ∪V = X. Then (X \U) is closed set and(X \V ) is regular closed set with (X \U)∩(X \V ) = ∅. By almost p-normalityof X, there exist disjoint preopen sets U1 and V1 such that X \ U ⊂ U1 andX \ V ⊂ V1. Let G = X \ U1 and H = X \ V1. Then G and H are preclosedsets such that G ⊂ U , H ⊂ V and G ∪H = X.

(b)⇒c and (c)⇒(a) are obvious.

One can prove that almost p-normality is also is regular closed hereditary.

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Almost p-normality does not imply almost p-regularity as the following ex-ample shows.

EXAMPLE 4.4 : Consider the space X as defined in Example 3.5. Then Xis almost p-normal, but it is not almost-p-regular since for the regular closedset {a, c} and the point b 6∈ {a, c}, there do not exist disjoint preopen setscontaining them.

However, we observe that every almost p-normal R0 space is a almost p-regular.

Next, we prove the following.

THEOREM 4.5 : Every almost regular strongly compact space X is al-most p-normal.

PROOF .This follows from Theorem 1.2 in [16], Corollary 3.1 [27] andTheorem 2.4 in [4].

Now, we state the invariance of almost p-normality in the following.

THEOREM 4.6 : If f : X → Y is continuous M-preopen rc-continuousand almost preirresolute surjection from an almost p-normal space X onto aspace Y, then Y is almost p-normal.

5. MILDLY P-NORMAL SPACES

DEFINITION 5.1 :A space X is said to be mildly p-normal if for everypair of disjoint regular closed sets F1 and F2 of X, there exist disjoint preopensets U and V such that F1 ⊂ U and F2 ⊂ V .

Clearly, every mildly normal space as well as almost p-normal space is mildlyp-normal.

We have the following characterization of mild p-normality.

THEOREM 5.2 : For a space X the following are equivalent :(a) X is mildly p-normal.(b)For every pair of regular open sets U and V whose union is X, there exist

preclosed sets G and H such that G ⊂ U , H ⊂ V and G ∪H = X.(c)For any regular closed set A and every regular open set B containing A,

there exists a preopen set U such that A ⊂ U ⊂ pCl(U) ⊂ B.(d)For every pair of disjoint regular closed sets, there exist preopen sets U

and V such that A ⊂ U , B ⊂ V and pCl(U) ∩ pCl(V ) = ∅.

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This theorem may be proved by using the arguments similar to those ofTheorem 4.3.

Also, we observe that mild p-normality is regular closed hereditary.

We define the following.

DEFINITION 5.3 : A space X is weakly p-regular if for each point xand a regular open set U containing {x}, there is a preopen set V such thatx ∈ V ⊂ V ⊂ U .

Clearly every almost p-regular space is weakly p-regular. The converse isnot true in general.For,

EXAMPLE 5.4 :Consider the space X as defined in Example 3.5. Then Xis weakly p-regular, but it is not almost p-regular since for the regular closedset {a, c} and the point b 6∈ {a, c}, there do not exist disjoint preopen setscontaining them.

However, it is clear that every weakly p-regular R0 space is almost p-regular.

It is obvious that every p1 -paracompact space is nearly paracompact.

Now, we prove the following.

THEOREM 5.5 :Every weakly p-regular p1-paracompact space is mildlyp-normal space.

PROOF .Let X be a weakly p-regular, p1-paracompact space and A, B beany pair of disjoint regular closed subsets of X. Since X is p1-paracompact, it issubmaximal and paracompact by Theorem1 in [7]. As X is submaximal weaklyp-regular it is weakly regular. So X is weakly regular nearly paracompactspace. Then by Theorem 8 in [29], X is mildly normal and hence mildlyp-normal.

In view of Theorem 5.5 we have the following.

COROLLARY 5.6 : Every almost p-regular p1-paracompact space ismildly p-normal.

Using Theorem 1 in [7], we can easily prove the following result.

THEOREM 5.7 : Every p1 -paracompact pre-T2 space is mildly p-normal.

Next, we prove the invariance of mild p-normality in the following.

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THEOREM 5.8 : If f : X → Y is an M-preopen rc-continuous and almostpreirresolute function from a mildly p-normal space X onto a space Y , thenY is mildly p-normal.

PROOF . Let A be a regular closed set and B be a regular open set contain-ing A. Then by rc-continuity of f , f−1(A) is a regular closed set contained inthe regular open set f−1(b). Since X is mildly p-normal, there exists a preopenset V such that f−1(A) ⊂ V ⊂ pCl(V ) ⊂ f−1(b) by Theorem 5.2.As f is M-preopen and an alomost preirresolute surjection,it follows that f(V ) ∈ PO(Y )and A ⊂ f(V ) ⊂ pCl(f(V )) ⊂ B.Hence Y is mildly p-normal.

THEOREM 5.9 :If f : X → Y is rc-continuous, M-preclosed functionfrom a mildly p-normal space X onto a space Y , then Y is mildly p-normal.

This theorem may be proved by using arguments similar to those in Theo-rem3.11.

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