Concerning p-closed topological spaces

              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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