Rev. Acad. Canar. Cienc., XIV (Núms. 1-2), 9-23 (2002)
Concerning p-closed Topological Spaces
M. N. Mukherjee, B. Roy and P. Sinha
Department o/ Pure Mathematics, University o/ Calcutta,
35, Ballygunge Circular Road, Kolkata-700019, India
e-mail: mukherjeemn@yahoo.co.in
Abstract
The present a.rticle aims at studying p-closed topological spaces, a sort of covering
property, introduced in [1) and studied further in [3, 7) in terms of preopen sets
and p-closure operator. We define pre-almost regula.rity and obtain sorne conditions
which follow necessa.rily in a p-closed space. These conditions turn out to be sufficient
too in the presence of the pre-almost regularity. Certain techniques via the
introduced concepta of p(8)-complete adherent point, p(8)- continuity of functions
and multifunctions , p(8)-compatible order, are employed to that end.
Keywords and phrases: p(8)-open sets, p(8)-closure, preopen set, pre-almost
regular, ¡rclosed space.
2000 AMS Subject Classification Code: 54D20, 54D99.
§1. Introduction
It is seen from literature, that a variety of compact-like covering axioms like
qua.si H-closedness [2], near compactness [8], S-closedness [9] etc. are being studied
quite meticulously by different mathematicians for a long time. In general, the
introduction and study of different types of open-like sets and closure-like operators
have given rise to the initiation of variant forms of covering properties. The
concept of p-closedness, yet again a type of covering axioms, is rather recent in
the hierarchy. In 1989, Abo-Khadra [l] proposed the concept of p-closed topological
spaces in terms of preopen sets of Mashhour et al. [6] . In a very recent
paper Dontchev et al. [4] continued with the investigation of such spaces. In [7],
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the notion of p-closedness was generalized for subsets of a topological space and
a long Jist of characterizations of such sets was obtained. Certain techniques via
sorne introduced concepts like p( B)-continuity, p( B)-su bclosedness and strong p( B)
closedness of graphs were employed there to obtain various characterizations of pclosed
spaces. The purpose of this paper is to continue the investigation of p-closed
spaces.
In the next section, we define p(B)-open and p(B)-closed sets. Then we introduce
the notion of pre-almost regularity- a concept which plays a pivota! role throughout
the paper. The rest of the paper is devoted to finding a number of characterizations
of pre-almost regular p-closed spaces. The appliances that we use to t his end are
p(B)-adherent point of sets, p(B)-continuity of functions and multifunctions, p(B)compatible
order etc, the terminologies being suitably defined in course of the
development of the paper.
Henceforth by spaces X, Y(or simply by X, Y) we mean topological spaces. A
set A in a space X is called preopen (preclosed) if A t;;; int el A, (resp. el int
A t;;; A), where c!A and intA respectively stand for the closure and the interior of
A in X. Clearly, A is preopen iff X \ A is preclosed in X . The class of ali preopen
sets of X will be denoted by PO(X), while PO(x) sh.all stand for the collection of
all preopen sets of X , each containing a given point x of X. For any A t;;; X, the
union (intersection) of ali preopen (preclosed ) sets in X, each contained in (containing)
A is called the preinterior (resp. preclosure) of A in X , denoted by pintA
(resp. pclA). Alternatively, pintA (pclA) can be defined as the largest (smallest)
preopen (preclosed) set contained in A (containing A). Again, it is known that
x E pclA(where A t;;; X and x E X ) iff for each U E PO(x), Un A ":f </J, and that
the relation connecting preclosure and interior is given by pcl( X\ A) = X \ pintA.
The above stated definitions, notations and results (and further relevant details)
are well known by now, and can be had, for instance, in [3] .
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§2. p( 8)-open set, pre-almost regularity and p-closedness
In [7] the pre-0-closure of a subset A in a topological space X, denoted by p(B)clA,
is defined to be the set of ali points x of X such that for each U E PO(x),
pclU n A ::f. </>. We now define as follows:
Deftnition 2.1. A subset A in a space X is called p(B)-closed if p(B)-c!A = A.
The complement of such a set is ca!led p(B)-open.
Remark 2.2. It follows from the above definition that a set A (~ X) is p(O)-open
iff for each x E A, there exists a U E PO(x) such that pclU <;;;; A. Also, clearly
every p(B)-open (p(B)-closed) set is preopen (resp. preclosed). That the converse
is false follows from the following example.
Example 2.3. Let X= {a,b,c} and T ={X,</>, {a}, {b}, {a, b}} Then (X, r) is a
topological space such that PO(X) =T. It is easy to check that pcl{a} = {a,c} ,
pcl{b} = {b, e} and pcl{a, b} = X. Now, A = {a, b} is a preopen set but there
<loes not exist any preopen set U containing x such that pc!U <;;;; A. Hence A is
not p(B)-open. Similarly, B = { b, e} is a preclosed set which is not p(B)-closed.
As is expected, in a regular space the collapse takes place, i.e.
Theorem 2.4. In a regular space X, a subset A is p(B)-open iff A is preopen.
Proof. lt suffices to show that a preclosed A in X is p(B)-closed. In fact , if x f/. A
then by regularity of X, there exists an open W such that x E W <;;;; clW <;;;;X\ A,
i.e. c!W n A=</>. Hence X f/. p(B)-clA.
Theorem 2.5. For any preopen set in a space X , pclA = p( B)-clA.
Proof. lt is clear that for any subset A of X, pclA <;;;; p(O)-clA. Next, !et A be
a preopen subset of X and x f/. pc!A. Then there exists V E PO(x) such that
V n A == </J. Then pc!V <;;;; pcl(X \A) == X \A so that pclV n A == </>. Hence
X f/_ p( 0)-clA.
The following result gives an expression for p( B)-closure of a set in a space X .
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Theorem 2.6 For any A~ X, p(B)-clA = n{ pclU: A~ U E PO(X ) }.
Proof. For any U E PO(X) with A~ U, we have p(B)-clA ~ p(B)-clU=pclU (by
Theorem 2.5.) so that L.H.S. ~ R.H.S .. Next, Jet x rf. p(B)-clA. Then for sorne
V E PO(x), pclV n A = r/>, so that A ~ X\ pclV = U (say)E PO(X) such that
pclU = X\pintpclV ~X \ V, and hence x rf_pclU. Thus x rf. R.H.S.
Definition 2. 7. For any subset A of a space X , we define the pre-ó-closure of A,
denoted by p(ó)-clA, to be the set of all x E X such that pint(pclU) n A =1- r/>, for
each U E PO(x).
Theorem 2.8. For any A ~ X, p(ó)-cl[p(ó)-clA) = p(ó)-clA.
Proof. That R.H.S . ~ L.H.S. is clear. Let x E L.H.S. and U E PO(x) . Then
pint(pc!U)np(ó)-clA =f. ef>. If y belongs to this intersection, then as pint(pclU)E
PO(y), we have (pintpclpintpclU)nA =f. ef>, i.e., pintpclU n A =1- ef>. Hence x E
R.H.S ..
Definition 2.9. A(~ X) is called pre-regular open if A =pintpclA . The complement
of such sets are called pre-regular closed . Thus A(~ X) is pre-regular closed
iff A =pcl(pintA)
Definition 2.10. A space X is called pre-almost regular if for each x E X and
each pre-regular open set V in X with x E V, there is a pre-regular open set U E X
such that x E U ~pclU ~ V.
We have seen in Theorem 2.8. that pre-ó-closure operator is an idempotent
one. But the next theorem which gives a few equivalent descriptions of pre-almost
regularity, incidentally shows that the p(B)-closure operator in a space becomes
idempotent iff the space is pre-almost regular.
Theorem 2.11. For a topological space X, the following are equivalent:
(a) X is pre-almost regular.
(b)For any A~ X , p(B)-clA = p(ó)-clA.
(c)For any A ~ X, p(B)-cl[p(B)-clA)=p(B)-clA.
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(d) For any A E PO(X), p(B)-cl[p(B)-clA]=p(B)-clA.
Proof. (a)=? {b): It is clear that p(ó)-clA ~ p(B)-clA. Let x E p(B)-clA and
U E PO(x). Then x Epint(pclU) . By (a), there exists a pre-regular open set V
containing x such that pclV ~pint(pclU). As x E p( B)-clA, we ha ve pclV n A f. ef>.
It then follows that pintpc!U n A 1= ef>.Thus X E p(ó)-clA.
{b)=? (e): For any A~ X, we have p(B)-cl[p(B)-clA]=p(B)-cl[p(ó)-clA][ by (b) ]=
p(ó)-cl[p(ó)-clA][ by(b) ]=p(ó)-clA. ( by Theorem 2.8. )
{e)=? (d): Obvious.
(d)=? (a): We first show that under the condition in (d), F = p(B)-clF, for
any pre-regular closed set Fin X ... ...... (l). Indeed, F being pre-regular closed,
F =pclU, where U=pintF is preopen. Now, p(B)-clF = p(B)-cl[pc!U]=p(B)-cl[p(B)c!
U] [by Theorem 2.5. ] =p(B)-clU [by (d) ]=pc!U [by theorem 2.5. ]=F. Now
to prove (a), !et x E X and U be a pre-regular open set containing x. Then
x tJ. (X\ U) = p(t'.1)-cl(X \U) [by (1), since (X\ U)is pre-regular closed ]. Thus
there exists V E PO(x) such that pclV n (X\ U) = ef>. Let W=pintpclV. Then
x E W and pc!W n (X \ U) =pclV n (X\ U) = ef>, i.e. pclW ~ U, where W is
pre-regular open set containing x. Hence X is pre-almost regular.
With the concepts developed so far, we now pass on to the study of p-closedness
of a topological space, the definition of the latter ccincept being introduced in [l]
as follows.
Definition 2.12 A topological space X is called p-closed if every cover U of X by
preopen sets of X has finite sub family U 0 such that X =U {pc!U : U E U 0 } .
The following result will be used in the seque!.
Lemma 2.13. In a p-closed space X, every family of p(B)-closed sets with finite
intersection property has non-void intersection.
Proof. Let U be a family of p( t'.1)-closed sets in a p-closed space X, having finite
intersection property. If íl U = r/>, then for each x E X, there exists Ux E U
such that x t/. Ux· Thus there exists Vx E PO(x) such that pcl Vx n Ux=ef>.
Then V = {Vx : x E X} is a cover of X by preopen sets, so that there exists
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a finite subset { x1, x2, .. ., Xn} of X with X = Ui::1 pc!Vx, . It then follows that
(íl?=i Ux,) íl(U~=l pclVx.) = </> and hence íl:=i Ux, = </>, contradicting the finite
intersection property of U.
Theorem 2.14. lf X is p-closed, then every cover of X by p(O)-open sets has a
finite subcover; the converse is also true if X is pre-alrnost regular.
Proof. Suppose U is a cover of a p-closed space X by p( 0)-open sets. For each
x E X, x E Ux for sorne Ux E U. Then pcl Vx <;;; Ux, for sorne Vx E PO(x) . As
{Vx : x E X} is a preopen cover of X by p-closedness of X, there exists a finite
subset {x1, X2, . .. , Xn} of X such that X= Ui::1 pclVx,· Then {Ux1 , Ux21 ., ., ., U:r:n}
is a finite subcover of U.
conversely, let X be pre-alrnost regular and U be a cover of X by pre-open sets
of X. Now, for each U E U, pintpclU is p(O)-open. In fact, as X is pre-alrnost
regular, x E pintpclU ::::} there exists V E PO(x) sucb that pclV <;;; pintpclU,
and hence pintpclU becornes p(O)-open. Thus {pintpc!U : U E U} is a cover of
X by p(O)-open sets. By hypothesis, there is a finite subset U 0 of U such that
X= U{pintpclU : U E U 0 } <;;; U{pc!U: U E U 0}, proving X to be p-closed.
Definition 2.15. A point x in a space X is called a p(O)-cornplete adherent point
of a set A <;;; X if for each p(O)-open set U containing x, IU n Al = IAI. where for
any B <;;;X, IBI stands for the cardinality of B.
Theorem 2.16. A pre-alrnost regular space X is p-closed iff every infinite subset
of X has a p( 0)-complete adherent point in X .
Proof. Let, if possible, A be an infinite subset of a p-closed space X, without any
p(O)-cornplete adherent point. Then to each x E X there corresponds a p(O)-open
set Ux containing x such that IUx n Al < IAl ....... (l) As {Ux : x E X} is a cover
of the p-closed space X by p(O)-open sets, by Theorem 2.14. there exists a finite
subset {x1,x21 .,., xm} of X such that X= U~ 1 Ux,· Then A= U~¡(Ux, nA) =
U~ 1 (Ux, n A) and hence IAI = max {IUx, n Al : i = 1, ... , m} which goes against
( 1).
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conversely, let the given condition hold in a space X which is pre-almost regular.
If X were not p-closed then by Theorem-2.14. there is a cover U of X by p(8)-open
sets such that for every finite sub collection U 0 of U, we have UU o C X .. .... (2)
Let k = min {\CI : C ~ U and C is a cover of X}. If No denotes the cardinal
number of the set of integers, it follows from (2) that k ? No. Now, there exists
W ~ U such that JU* I = k and u· is a cover of X . We can well order u· by sorne
well-ordering ~ - Let U E u• be arbitrary. Since ~ is a minimal well-ordering, we
have J{V E u• : V~ U}\ ~ \U*\. Also by definition of k, {V E u• : V~ U} is not
a cover X, for any U E U*. For each U E U*, we can choose inductively a point
Xu E X \ U{ { xv} u V : V E u· and V ~ U} and define a set A = { Xu : u E u·}.
It then follows at once that (U"::/; V'* xu "::/; xv). Hence \A\ = k ? No. It now
suffices to show that A has no p( 8)-complete adherent point in X. In fact, let
y E X . As U* is a cover of X, there exists U* E U* such that y E U*, where U*
is p(8)-open. Again, for U E U* with xu E U* we have by choice of xu , U ~ U* .
Hence {U E U* : xu E U*} ~ {U E U* : U ~ U*}. But by minimality of ~,
\{U E u· : u ~ U*}I < k and hence \A n u·¡ < k = IAI. Thus y cannot be a
p( 8)-complete adherent point of A and this completes the proof.
Definition 2.17. [7], (a) A net {x0 : a E (D, ?)} (where (D, ?) is a directed set)
in a topological space (X, r) is said to p( 8)-adhere at sorne point x E X, written
as x E p(8)-ad(x0 ), if for each preopen set U containing x and each a E D, there
exists a /3 E D with f3? a such that XfJ E pclU.
(b)A filterbase :Fon a space X is said to p(8)-adhere at sorne point x E X, written
as x E p(8)-ad:F, if for each U E PO(x) and each FE :F, F n pclU "::/; rjJ.
Sorne of the characterizations of p-closed spaces, obtained in [7], are quoted
below to be used in the sequel.
Theorem 2.18. For a space X, the following are equivalent:
(a)X is a p-closed space.
(b)Every filterbase on X p(8)-adheres in X .
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(c)Every net in X p(O)-adheres in X.
Theorem 2.19. A pre-almost regular space X is p--closed iff every net in X
with a well-ordered index set as domain, p( 0)-adheres in X.
Proof. The necessity part of the theorem follows at once from Theorem 2.18.
For the converse, let A be an infinite set in X. Then A can be well-ordered and it
can be considered to be a net with itself as its domain. Let this net p(O)-adhere at
a point x E X. It is then easy to see that any U E PO(x), IAnpclUI = IAI. Now,
for any p(O)-open set V containing x, there exists U E PO(x) such that pclU ~V,
so that IA n VI = IA n pclUI = IAI. hence x is a p(O)-complete adherent of A and
then X becomes p-closed in view of Theorem 2.16.
§3. p( 8)-continuous functions and multifunctions, p( 8)-compatible order
and p-closedness.
Let X, Y be topological spaces. By a multifunction F from X to Y, written
as F : X --+ Y, we shall mean, as usual, a function which takes points of X into
nonempty subsets of Y.
Defl.nition 3.1. A multifunction F: X --+Y, where X, Y are topological spaces,
is called upper (lower) p(O)-continuous if for each U E PO(Y), F+(U) (resp.
F-(U)) is p(O)-open in X, where for any A~ Y, F+(A) = {x E X : F(x) i; A}
and F-(A) = {x E X: F(x) n A ;f:. </>}
In the next two results we give sorne equivalent ways of defining upper and
lower p(O)-continuity of multifunctions.
Theorem 3.2. For any multifunction F: X --+Y, the following are equivalent:
(a) F is upper p(O)-continuous on X.
(b) For each x E X and each preopen set V containing F(x), there is a p(O)-open
set U in X containin.g x such that F(U) ~ V.
(c) For each x E X and each preopen set V containing F(x), there exists U E
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PO(x) such that F(pclU) e;;;; V.
(d) For each preclosed set V in Y, F - (v) is p(O)-closed in X .
Proof. (a)~(b): Let x E X and V E PO(F(x)). Then by (a), F+(V) =U (say)
is a p(O)-open in X . Also, x E U (as F(x) e;;;; V) and F(U) e;;;; V.
(b)~ (a) : Let V be a pre-open set in Y and x E F+(V). Then F(x) e;;;; V. By (b),
there is a p(O)-open set U in X containing x such that F(U) e;;;; V. U being p'(O)open
containing x, there is a preoopen set W in X su ch that x E W e;;;; pcl W e;;;; U.
Then F(pclW) e;;;; F(U) e;;;; V, i.e., pclW e;;;; F+(V), proving that F+(V) is p(B)open
in X . Hence (a)follows.
(a)~ (e): Let x E X and V E PO(F(x)). Since F+(V) is p(O)-open and x E
F+(v), there exists U E PO(x) such that pclU e;;;; F+(V) , i.e., F(pclU) e;;;; V.
(e)~ (a) : Let V E PO(Y) and x0 E F+(V) . By (e), there is U E PO(x0) such
that F(pclU) e;;;; V, i.e., pclU e;;;; F+(V) . Hence F+(V) is p(O)-open in X .
(a)~(d) : This is clear from the known facts that for any Be;;;; Y, X\ F+(B) =
F-(Y \ B), X \ F-(B) = F+(Y \ B) and that a subset A of X is p(O)-open iff
X\ A is p(O)-closed.
Theorem 3.3. For a multifunction F : X -+ Y, the following are equivalent:
(a) F is lower p(B)-continuous on X.
(b) For each x E X and each preopen set V in Y with x E F-(V), there exists a
preopen set U in X such that x E U e;;;; pclU e;;;; F-(V) .
(e) For each x E X and each preopen set V in Y with V n F(x) # </>, there exista
U E PO(x) such that x0 E pclU ~ F(x0 ) n V#</>.
(d) For each preclosed set B in Y, F+(B) is p(O)-closed in X.
Proof. The proof, being similar to that of Theorem 3.2., is omitted.
In the next theorem the concept of p(B)-continuity of multifunction is used to
the study of p-closed spaces.
Theorem 3.4. In a pre-almost regular space X, the following conditions are equivalent(
where the implications (a)~ (b) ~(e) always hold irrespective of whether X
is pre-almost regular or not) .
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(a) X is p-closed.
(b) For each upper p(B)-continuous multifunction Fon X to any topological space
Y, the multifunction F* on X to Y, defined by F*(x) = p(B)-cl F(x) (for x E X)
assumes a maxirnal value under set inclusion.
(e) Ea.en upper p(O)-continuous rnultifunction Fon X to any topological space Y
with p(B)-closed point images assurnes a maxmal value under set inclusion.
Proof. (a)=> (b): It is to be shown that the set Z = {F*(x) : x E X},
partially ordered by inclusion has a maximal element. Consider a linearly ordered
subset L in Z. For each y E X, such that F*(y) E L, consider the set
B(y) = {x E X : F*(y) ~ F*(x)}. As L is a chain, B = {B(y) : y E X and
F*(y) EL} is a filterbase on X and hence by (a) and Theorern 2.18., there exists
x• E p(B)-ad B. Let B(y) E B. We show that B(y) is p(B)-closed. Let z E p(B)clB(
y) and Jet W be a preopen set containing F(z). Then by Theorem 3.2., there
is a p(B)-open set U in X containing z such that F(U) ~ W. Then for sorne
V E PO(z), pclV ~ U and hence F(pclV) ~ F(U) ~ W. As z E p(B)-clB(y),
there exists x0 E pclV n B(y). Now, x0 E B(y) => F*(y) ~ F*(x0) = p(B)c1F(
x0) ~ p(B)-cl F(pc!V) ~ p(B)-clW=pclW (using Theorern 2.5.). Thus for
every preopen set W containing F(z), F*(y) ~ pclW. It then follows by Theorern
2.6. that F*(y) ~ p(B)-clF(z) = F*(z) so that z E B(y). Hence B(y) becomes
p(B)-closed. Now, x• E íl{p(B)-clB: BE B} = ílB and hence F*(x*) is an upper
bound of L. Thus by Zorn's lemma, Z assumes a maximal value under set inclusion.
(b)=>(c): This is obvious.
(c)=>(a): Suppose that X is pre-alrnost regular, satisfies (e) but is not p-closed.
Then by Theorern 2.19., there is a net T = {T0 : a E D} having no p(B)-adherent
point in X, where D, the dornain of T, is a well-ordered set. Let D be given the order
topology. For each a E D,we define the set U0 = X\p(B)-cl{T;, : >.~a}. As X
is pre-almost regular, U0 is p(B)-open by Theorern 2.11. and hence preopen. Again,
for any x E X, x is not a p( B)-adherent point of T and hence for sorne U E PO ( x)
and sorne a E D, pclU n {T>. : >. ~ a} = rj>, i.e., x tf_ p(B)-cl {T>. : >. ~ a} so that
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x E U0 • Thus {U0 : a E D} is a preopen cover of X. For each x E X, there is a first
suffix a(x) (say) in D such that x E Uo(z)· We define a multifunction F: X-+ D
by F(x) = {>.E D : >. :::; a(x)} . Since D is endowed with the arder topology which
is regular, each F(x) (for x E X) is closed so that preclosed and hence p(B)-closed
(by Theorem 2.4.) . We claim that F is upper p(O)-continuous. For this !et x E X
and let W be a preopen set in D containing F(x) . Now, Uo(z) is a p(O)-open set
in X containing x, so that it suffices to show by virtue of Theorem 3.2 [(b} => (a)]
that F(Uo(z)) ~ W . Indeed,y E Uo(z) => a(y) :::; a(x) => F(y) :::; F(x) ~ W . Thus
F(Uo(zJ) ~ W and F becomes upper p(8)-continuous. To complete the proof, it
is only to be proved that F cannot assume any maximal value with respect to
set inclusion relatfon. If possible, suppose that for sorne x0 E X , F(x0 ) is maximal.
Then for any y E X, F(y) $ F(x0) so that Uo(y) ~ Uet(:z:o ) and consequently
y E Uo(zo)· Hence X= Uo(zo) which is absurd.
We now introduce two other definitions and pass on to characterize p-closedness
with these concepts as the supporting appliances.
Definition 3.5. A pre-arder relation :::; (i.e.,$ is reflexive and transitive) on a
topological space X is said to be lower (uppér) p(O)-compatible if the set {y E X :
y:::; x} (resp. the set {y E X : x:::; y}) is p(O)-closed, for each x E X .
Definition 3.6. A function f from a topological space X to a partially arder set
(poset) (Y, $) is called lower (upper) p(8)-continuous if ¡-1( {y E Y : y :::; y"} )
(resp. ¡-1 ({y E Y : y• $ y})) is p( 0)-closed in X for each y• E Y .
Theorem 3.7. For a pre-almost regular space X, the following are equivalent (the
implications (a) => (b) => (e) => (d) being always true irrespective of whether the
space X is pre-almost regular or not):
(a) X is p-closed.
(b) X has a maximal element with respect to each upper p(8)-compatible pre-arder
relation on X.
(c) Each upper p(O)-continuous function f : X -+ Z, where Z is a poset, assumes
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a maximal value.
(d) Each upper p(8)-continuous multifunction F from X to a T1-space Y assumes
a maximal value w.r.t. set inclusion relation.
Proof. (a)~ (b): Let X be a p-closed space in which an upper p(8)-compatible
pre-order relation ~ is defined, and let '!/; be a linearly ordered subset of (X,~).
For each x E X, the set 'l/Jx = {y E X : y ~ x} is p( 8)-closed (by hypothesis). As
'!/; is a chain, the family { 'l/Jx : x E '!/;} has finite intersection property. By Lemma
2.13., there exists x0 E íl{ 'l/Jx : x E '!/; }. Then x0 ~ x, for each x E '!/;, i.e., xo is
an upper bound of '!/;. By Zorn's Lemma it then follows that X has a maximal
element.
(b)~(c): Let f : X ~ Z be an upper p(8)-continuous function , where (Z, :S)
is a given poset. Let us define a relation ~ on X as : for x, y E X, x :S y
in X iff f(x) :S f(y) in Z. Then (X,~) is obviously a pre-ordered set. Now,
for each x E X, ¡-1({z E Z : f(x) :S z}) = {y E X : x :S y} = A(x) (say)
which is p(8)-closed in X, since f is upper p(8)-continuous. Hence ':S ' on X is an
upper p(8)-compatible pre-order relation. By (b), it then follows that (X, :S) has
a maximal element x0 (say). Then the definition of :Son X shows that f assumes
a maximal value at x0 of X .
(e)~ {d): Let F: X~ Y, be an upper p(8)-continuous multifunction from X to
a T1-space Y. We can treat F to a function from X into the poset ('P(Y) \ { <P }, ~),
where 'P(Y) denotes the power set of Y, and show that this function F is upper
p(8)-continuous. So let C0 E 'P(Y) \ {</>}. It is to be shown that F - 1{C E
'P(Y) \ { </>} : C0 ~ C} is p( 8)-closed in X . For this, suppose x í/. p-1 ( { C E
'P(Y) \ { <P} : Co ~ C} ). Then F(x) "/; C, for any C( E 'P(Y) \ { <P}) 2 Co, so
that there exists Yo E C0 \ F(x). Then F(x) ~ Co \ {Yo} . Now, F-({Yo}) is
p(O)-closed in X, as Y is T1 and F is an upper p(8)-continuous multifunction
(see Theorem 3.2.); also, x íf. p- ( {y0 }) . Thus there exists a preopen set U with
x E U such that pclU n F-({y0}) = </>, i.e., y0 E Y\ F(pclU). Consequently,
el Un p-l ( {e E 'P(Y) \ { <P} : e 2 Co}) = <P and hence X cannot be a p( 8)-adherent
point of p-1({C E 'P(Y) \ {</>} : C 2 C0}) which then becomes p(8)-closed. The
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rest follows from (e).
(d)=> (a): Follows similarly as in the proof of (e)=> (a) of Theorem 3.4.
Theorem 3.8. For any topological space X, the implications (a) => (b) => (e)
hold, while '(d) => (a)' is also true if X is pre-almost regular, where the statements
(a) to (d) are as follows.
(a) X is p-closed.
(b) X has a minimal element with respect to each lower p(B)-compatible pre-order
relacion on X.
(e) Each lower p(B)-continuous function from X to a poset Z assumes a minimal
value.
(d) Each lower p(B)-continuous multifunction F from X into the preclosed subsets
of a space Y assumes a minimal value with respect to set ínclusion relation.
Proof. The proofs of '(a)=> (b)' and '(b)=>(c)' are quite similar to those of Theorem
3.7.So let us prove '(c)=>(d)'and '(d)=> (a)' as follows.
(c)=>{d): Proceeding similarly as in the proof of '(e)=> (d)' of Theorem 3.7,
let the given multifunction F be regarded as a function from X to the poset
(F*, ~),where F* is the collection of all nonempty preclosed sets in Y, and let us
first prove that this function is lower p(B)-continuous. For this Jet C0 E F* and
x r/. F-1({C E F* : C ~ Co}). Then CE F* and C ~ Co => F(x) f. C, i.e.,
F(x) n (Y\ C0) f. c,P ....... (l) Now, by lower P(B)-continuity of the muhifunction
F : X --+ Y, X\ F-(Y \ C0 ) is p(B)-closed in X ( as Y\ C0 E PO(Y) ), and
by (1) it follows that x ~ X \ F-(Y \ C0). Thus there exists U E PO(x) sucb
that pclU n [X\ F- (Y \ C0)] = ef>, i.e., pclU ~ F-(Y \ C0). This shows that
z E pclU => F(z) n (Y\ C0) f. <P => F(z) \ C0 i- <P => F(z) i- C, for any C E ;:•
with C ~ C0 . Thus pcJUnF- 1({C E F* : C ~ C0}) = ef> which proves that
x ~ p(B)-cl({C E ;:• : C ~ C0 }) establishing that F-1({C E F* : C ~ Co})
is p(B)-closed in X and hence that F, treated as a function from X to the poset
(F*, ~),is lower p(B)-continuous. Then by (e), F assumes a minimal value with
respect to the set-inclusion relation.
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(d)~(a): We follow the proof of Theorem 3.4. ((e)~ (a)) . Let X be a prealmost
regular space which is not p-closed. Then by Theorem 2.19., there is a
net S = {S0 : a E (D, 2:'.)} with a well-ordered set (D, 2:'.) as its domain such
that S has no p(B)-adherent point in X . Let D be given the order topology. We
put Vi. = X\ p(B)-cl( {S0 : a 2:'. >.}) for each >. E D. Then proceeding similarly
as in the proof of '(e)~ (a) ' of Theorem 3.4., {Vi. : >. E D} is preopen
cover of X . Then as before, we define a multifunction F : X -+ D, given by
F(x) = {a E D : a 2:'. >.(x)}, where >.(x) is the first element in D such that
x E Vi.(x) . The arguments similar to the corresponding ones in the proof of Theorem
3.4. ((e)~ (a)) and use of Theorem 3.3. yield that F is a Jower p(B)-continuous
multifunction from X into the preclosed subsets of the space Y , assuming no minimal
value with respect to set inclusion relation.
We put an end to our deliberation after showing an interesting application of
the results, done so far. It is a fixed set theorem for any multifunction on a p-closed
space into itself, stated as follows:
Theorem 3.9. Let X be a p-closed space. Then every multifunction F from X
into itself, carrying p(B)-closed sets into p(B)-closed subsets of X , fixes a non-void
p(B)-closed set, i.e. there exists a non-void p(B)-closed subset A of X such that
F(A) =A.
Proof. Let U= {A~ X: A":/;</>, A is p(B)-closed and F(A) ~A} . Clearly X E U.
We shall apply Zorn's lemma on U which is a partially ordered set under inclusion
'~ '. Let 'l/J be a linearly ordered su bset of U. Pu t B = n'¡/J. As 'l/J is a family of p( 0)closed
set with finite intersection property and X is p-closed, we have by Lemma
2.13., B ":/; </>. Also, intersection of any family of p( B)-closed sets can be shown to
be p(B)-closed. Moreover, F(B) ~ F(A) ~ A, for each A E 'l/J ~ F(B) ~ B. Thus
Bis a lower bound of 'ljJ and BE U. By Zorn's lemma, U has a mínima! element
A ( say ). Now, as A(# </>) is p(B)-closed, by hypothesis F(A) is p(B)-closed and
F(F(A)) ~ F(A) ( since A E U, F(A) ~ A ). Hence F(A) E U with F(A) ~ A.
By minimality of A, we conclude that F(A) = A.
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