p-Closedness in topological spaces

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Rev. Acad. Canar. Cienc., XII (Núms. 1-2), 33-50 (2000) - (Publicado en julio de 2001)
p-CLOSEDNESS IN TOPOLOGICAL SPACES
M. N. MUKHERJEE, ATASI DEBRAY ANO P. SINHA
ABSTRACT The present paper deals with a kind of covering property for topological
spaces, called p-closedness, first initiated by Abu-Khadra [4], followed by a recent study ofthe
samc by Dontchev et al. [3]. We derive here a number of characterizations and certain relevant
prc•ptrties of such a concept mainly via certain newly introduced notions like p(S)-continuity,
p(lJ )-subclosedness and strong p(8)-closedness of graphs of functions .
. KEYWORDS Preopen set, preclosure, p-closed space, p(8)-continuity, p(S)-subclosed
graph, stron~Iy p(S)-closed graph.
1995 AMS SUBJECT CLASSIFICATION CODE 54020, 54C50
§ 1. INTRODUCTION
Compactness and its different allied forros, specially quasi H-closedness, have so far
been studied in detail by many topologists. It is seen from literature that certain open-like sets,
e.g. semiopen sets [5], regular open sets, have been employed for the above investgations. The
notion of preopen sets was initiated by Mashhour et al. [7] and such sets along with sorne
other íelevant concepts have been investigated by many. Recently, Dontchev et al. [3] have
takcn up an investigation of a sort of covering property, called p-closedncss for topological
spaces with the help of the notions of preopen sets and sorne associated oncs, the originator of
the notion of p-closedness being Abo-Khadra [4] in 1989. We propose to undertake, in this
paper, a further study ofthe same.
In [3], certain characterizations of p-closedness for topological spaces and their subset~
have been found mainly in terms of filterbases. We shall add a few more to this list of
formulations, in section 2.
In section 3, we define a type of functions called p(S)-continuous ones, and introduce
the notion of p(S)-subclosedness of graphs of fünctions. These ideas are exploited to
ultimately achieve a few characterizations of p-closed topological spaces.
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In section 4, we try to obtain formulations of p-closed spaces by means of strong p(8)­closedness
of graphs of functions, a concept suitably defined in the same section.
In what follows, by spaces X and Y we mean topological spaces. For any subset A of a
space X, we shall use the notations clA and intA to denote the closure and interior of A in X
respectively. The word 'neighbourhood' will be abbreviated as ' nbd' .
§ 2. p-CLOSEDNESS. IN GENERAL
Let us, at the very outset, clarify certain key words and notations that are often taken
resort to throughout the paper. These known terminologies and other pertinent details can be
found in literature (e. g. see [2])
DEFINITION 2.1 : A subset A of a topological space X is called preopen if A ~ intclA, the
complements of such sets in X are known as preclosed .\·ets i.e., for a preclosed set A, clintA ~ A.
The collection of all preopen sets in a space X will be denoted by PO(X).
DEFINITION 2.2 : For any space X and any A ~ 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, to be denoted by pintA (resp pclA).
Since arbitrary union (intersection) of preopen (preclosed) sets in X is known to be preopen
(preclosed), the preinterior (preclosure) of a set A in a space X can equivalently be defined as
the largest (smallest) preopen (preclosed) set contained in A (containing A). The following
results are also well-known.
THEOREM 2.3 : Let A be a subset of a space X. Then,
(a) pclA consists precisely ofthose points x ofX such that UnA :t: <!>. for every preopen set U
containing x ;
(b) pcl(X \ A) = X \ pintA.
We now append the definition ofp-closedness. as introduced originally in [4l
DEFINITION 2.4 : A non-void subset A of a topological space X is said to be p-cloxed
relative to X if for every cover {U., . a E /\} ¡ here and hereafter J\ denotes an indexing set) of
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A by preopen sets ofX, there exists a finite subset Ao of A such that A~ u{pclUa: a E A0 } .
If, in addition, A = X, then X is called a p-closetl space.
THEOREM 2.5 : Suppose A and Y are subsets of a topological space X such that A ~ Y ~
X and Y is open in X. Then A is p-closed relative to X if and only if A is p-closed relative to
the subspace Y.
PROOF: First suppose that A is p-closed relative to Y, and !et {Va : a E J\.} be a cover of A
by preopen sets of X. As Ya E PO(X) and Y is open in X, one can check that Ya n Y E PO(Y).
Hence, { Ya n Y : a E A} is a cover of A by preopen sets of Y. By p-closedness of A relative
to Y, there exists sorne finite subset Ao of A such that A~ u{pclv (Va nY): a E A0} (here
and afterwards also, the usual notation pclv B stands for the preclosure of a subset B of Y in
the subspace Y ofX). It is then easily seen that A~ u{pclVa : a E Ao} , proving that A is p­closed
relative to X.
Conversely, Jet A be p-closed relative to X and { Va : a E J\.} be a cover of A by preopen sets
ofY. As Yis open, and Ya E PO(Y) for each a E J\., we have Ya E PO(X). Thus, {Va : a E J\.}
is a cover of A by preopen sets of X. By hypothesis, there is a finite subset J\.0 of J\. such that
A~ u{pclVa : a E Ao}, i.e., A n Y= A~ u{(pclVa) n Y: a E Ao}. It can be verified that
pclVa n Y ~ pclv Vª . Thus, A ~ u { pclv (Vª) : a E J\.o}, and A becomes p-closed relative to Y.
COROLLARY 2.6: An open subset A ofa space X is p-closed iffit is p-closed relative to X.
THEOREM 2.7: (a) Union offinite number ofsets in a space X, each ofwhich is p-closed
relative to X, is p-closed relative to X.
(b) A subset A in a p-closed space X is a p-closed relative to X if A is preopen as well as
preclosed.
PROOF : The straightforward proofs are omitted.
It is well-known that barring paracompactness the best known weaker forro of
compactness is quasi H-closedness, a few other widely studied compact-like covering
properties being S-closedness [9], s-closedness [6] and near compactness [8]. A topological
space X is called quasi H-closed [ 1 l if every open cover '1i of X has a finite subfamily . o/lo, the
mion of the closures of whose members is X. As to the relation of p-closedness of a space
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with the above standard covering properties, it has been shown by Dontchev et al. [3] that p­closedness
of a space X is independent of each of compactness, near compactness, s­closedness
and S-closedness of X. Now, since every open set is preopen and for any open set A
in X, pclA = clA. it immediately follows that
THEOREM 2.8 : Every p-closed space is quasi H-closed.
The above result and also the fact that the converse of it is false, have also been
observed in [3). We give here another anda rather simple example of a compact space which
is not p-closed.
EXAMPLE 2.9 : Let X denote the set of real numbers endowed with the cofinite topology.
The space is clearly compact. Now, for any subset A of X, there are three possibilities as
follows : (i) A is finite, (ii) A is infinite with finite complement, (iii) A is infinite with X \ A
infinite.
For case (i), intclA = intA = ~. so that A is not preopen. In case (ii), A is open and hence preopen.
For case (iii), we have A~ X= intclA and in this case A is preopen. Let U;= (X\ N) u {i}, for
any i EN (the set ofnatural numbers). Thus t¡,/= {U; : i EN} u {(X\ N)} is a cover ofX by
preopen sets of X. We observe that X \ N and X \ U; ( i E N) are also preopen (being sets of
type (iii)). Hence there cannot exist any finite subfamily of 411, the preclosure of whose
members may cover X. Thus, X is not p-closed.
Similar to the definition of 9-adherence and 9-convergence of filterbases, the ideas of
pre-9-adherence and pre-9-convergence of filterbases were introduced in [2]. We shall recall
them below, and include with them the corresponding definitions for'flCtS.
DEFINITION 2.10 : Let X be a topological space, A~ X and x E X.
(a) [3] Afilterbase $ on A is said to
(i) pre-0..adhere at x (to be written as x E p(9)-adffe}, if for each preopen set U containing x
and each F E $, F n pclU ~ ~.
(ii) pre-0..converge to x (to be denoted by$~ x) if for each preopen set U containing
X, there exists F E sr such that F ~ pclU.
(b) A net {Xa: ce E(D,2:)}(where (D . ~) is directed set) in A is said to
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(i) pre-()..adhere at x (written as x E p(9)-ad(x,.)), if for each preopen set U containing x and
each a E D, there exists a P E D with P ~a such that Xp E pclU;
(ii) pre-()..converge to x ( in notation, Xa ~ x) if the net is eventually in pclU, for each
preopen set U containing x. lf a net {Xa} pre-9-adheres at sorne x E X (pre-8-converges to
sorne x E X), we shall sometimes write that x is a pre-()..adherent point o/ the net (resp. the
net is pre-()..convergent to x). Similar terrninologies apply to filterbases also.
DEFINITION 2.11 : For a subset A of a space X and a point x of X, we say that x is in the pre-()..
closure of A and writex E p(@-clA, iffor every preopen set U containing x, pclU 11 A~<!>.
The rest of this section is forwarded to the characterizations of p-closedness in terms of
the concepts detailed so far. The first theorem in this endeavour follows next, giving a long list
of formulations of p-closedness of subsets relative to a space, the first three being already
known from Dontchev et al. [3).
THEOREM 2.12 : For any non-void subset A of a space X, the following statements are
equivalent :
(a) A is p-closed relative to X.
(b) Every maximal filterbase on X which meets A, pre-0-converges to sorne points of A.
(e) Every filterbase on X which meets A, pre-9-adheres at sorne point of A.
(d) For every farnily {Ua: a E A} ofnon-void preclosed sets with (nuª) 11A = <!>. there is
e.e/\
a finite subset Ao of A such that ( n p int U (1) 11 A = <I> .
ClEAo
(e) Every maximal filterbase on A is pre-9-convergent to sorne point of A.
(t) Every filterbase on A is pre-9-adheres to sorne point of A.
(g) Fo' every faallly {B. : a e A} of non-empty ""' in X Mth [O(p(0) - dB.)] r> A = + ,
the<e •>dsts •finito sub8"1 A, of A suoh t1"t (0,s.) r>A =O.
(h) Every net in A pre-9-adheres at sorne point of A
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(i) Every ultranet in A pre-0-converges to sorne point of A.
(j) Every net in A has a pre-0-convergent subnet.
PROOF: The equivalence of(a), (b), (e), (d) has been established in [3], and '(e)=> (f)' is
obvious. We prove the other implications as follows.
(b) ~(e): If .9íis a maximal filterbase on X such that .9ímeets A, then $* = {FnA: F E $}is
a maximal filterbase on A. Hence '(b) =>(e)' is clear. The converse is obvious.
<O => (g) : Let $ = {B .. : ex E A} be a family of non-void sets in X such that for every finite
subset Ao of A, [o B.) ,-, A ' ~ . Thon ll' = <[o B.) ,-, A ' A, ;,. finite •ubset of A} ;,.
filterbase on A. By (f), let a E A n (p(0)-ad.91). Then for each ex E A and each preopen set U
containing a, AnB..n(pclU) ~ cjl, i.e., B .. npclU ~ cjl. Hence a E p(O)-clB .. for each ex E A and
consequently, (O p(0)- clB11) n A~ cjl .
<g> =>(a): Let {U .. : ex e A}be a cover of A by preopen sets ofX. Then A n [íl(x \u. . )]=cjl .
a eA
Iffor sorne ex E A, X\ pclU .. = cjl, then we are through. lf(X \ pclU .. )(= B.,. say) ~ cjl for all ex E A,
then $ = {Ba : ex E A} is a farnily ofnonempty sets such that
(1)
In fact, let x E p(O)-clBa = p(O)-cl(X \ pclUa).Then for every preopen set v. containing x,
(X\ pclUa) n pclV. ~ cjl. Since U11 E PO(X), ifx E U .. then (X\ pclU .. ) n (pclUa) ~ cjl which is
not possible. Thus x ~ Ua so that x E X\ U ... Hence p(O)-clBa ~ X\ U .. and (i) follows. By (g),
thondu futlte wbset Ao of A "'ch that [o B.) ,-, A = ~;.e., A <: X 1D.(X1 pc!U • ) = !d pe! U.
which proves (a).
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(Q <::> (h) : Suppose (f) holds. Let { x,, : n e(D, ~)}be a net in A. Consider the filterbase
1f:= {T. : ne D}generated by the net, where T0 = {xm : m E D and m ~ n} . By (f), there exists
a E A n (p(8)-ad~. Then for each preopen set U containing a and each F E §, pclU n F ".f:. <!> i.e.,
pclU n T0 ".f:. <!>. for ali n E D. Hence a E A n [p(8)-ad(x.)], and (h) follows. The proof of
'(h) => (f)' is quite similar.
(h) => (i) : Let { x0 : n e(D, ~ ) }be an ultranet in A. By (h), there exists a E [p(8)-ad(x0)] n A.
Let U be a preopen set containing a. Since the given net is an ultranet in A, it is eventually in either
(A n pclU) or A\ (A n pclU). But since the net is frequently in (A n pclU) (as a E p(8)-ad(x0 )),
we conclude that the net is eventually in pclU. Hence x,, ~ a.
(i) => (j): We know that every net in A has a subnet which is an ultranet. Thus any given net
in A has a subnet which pre-8-converges to sorne point of A (by (i)), and (j) follows.
(j) => (h) : Let T : E ~ A be a pre-0-convergent subnet of a given net S : D ~ A, and
suppose T ~ a E A. Then T = Sog, where g : E ~ D is a function such that for each n E D
there exists p E E with the property tbat g(m) ~nin D whenever m E E with m ~ p. Let U be a preopen
set containing a and n E D. There is m1 E E such tbat T(m) E pclU, for all m ~ m1(m E E). For the
given n E D, Jet p E E with the above stated property and Jet m2 E E such that m2 ~ p, m1.
Then g(m2) ~ n in D, and we have T(m2) = S0 g(m2) E pclU(since m2 ~ m1) . Hence a E
(p(0)-adS) n A. This completes the proof of the theorem.
Putting A = X in the above theorem, we obtain the following characterizations of p­closed
spaces (we note here that direct proofs ofthe equivalence ofthe statements in (a) - (d)
ofthe following theorem have been given by Dontchev et al. (3)).
THEOREM 2.13 : For a space X, the fóllowing are equivalent :
(a) X is a p-closed space.
(b) Every maximal filterbase on X pre-0-converges.
(e) Every filterbase on A is pre-0-adheres to sorne point of A.
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( d) For every family { Ua : a E A} of non-void preclosed sets in X with n u a = cj> , there is a
a eA
finite subset Au of A su ch that n p int U 11 = cj> .
a EA o
(e) For every farnily {Ba : a E A} of non-ernpty sets in X withíl p(8) - clBª = cj>, there
exists a finite subset Ao of A such that n Bª =el>.
a eA0
(f) Every net in X pre-8-adheres at sorne point ofX.
(g) Every ultranet in X, pre-8-converges.
(h) Every net in X has a pre-8-convergent subnet.
ª '"
TH EOREM 2.14 : A space X is p-closed if and only if every filterbase on X with at most one
pre-8-adherent point pre-8-converges.
PROOF : Let X be p-closed and $'be a filterbase on X with at most one pre-8-adherent
point. By theorern 2.13, $'has then a unique pre-8-adherent point -xo (say) in X. Let §do not
pre-8-converge to Xo. Then for sorne preopen set U containing Xo, and for each F E §,
F 11 (X\ pclU) ~ cj>. So, rJ = {F 11 (X\ pclU) : F E §}is a filterbase on X and hence has a pre-
8-adherent point x in X. Since U is a preopen set containing Xo such that (pclU) 11 G = el>. for
ali G E <fJ, we have x ~ Xo. Now, for each preopen set V containing x and each F E §, F 11
(pclV) ;¿ F 11 pclV 11 (X\ pclU) ~ el> i.e., F 11 (pclV) ~ cj>. Thus, x is a pre-8-adherent point of
§, where x ~ Xo. This contradicts that Xo is the only pre-8-adherent point of $. The converse is
clear in view of Theorern 2.13 ((e) ~ (a)) and the fact that a point x is necessarily a pre-8-
adherent point of a filterbase § if $' ~ x.
DEFINPl.ON 2.15 : A family q¡ of preclosed subsets of a space X will be called a precaver
of X if for each x E X, there is sorne U E q¡ such that U is a prenbd of x (i. e., x E V e U), for
sorne preopen set V).
THEOREM 2.16 : A space X is p-closed iff every precover of X has a finite subcover.
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PROOF : Let X be p-closed and ó{t be any precover of X. Then, for each each x E X, there
are U, E 41/ and a preopen set V, such that x E V, e U •. It then follows that {V. : x E X } is a
n
cover of X by preopen sets of X. By p-closedness of X, X = LJ pclV,, , for a finite subset
t=I
n
{ x 1, ... , x0 } of X. Thus, X = LJ U•; . Conversely, if U is any preopen cover of X, then { pcl U :
,,,J
U E 11(/ }is precover ofX and hence the rest follows trivially.
§ 3. p(0)-CONTINUITY. p(0)-SUBCLOSED GRAPH ANO STRONG p(0)-CLOSEDNESS
According to our proposed scherne, we introduce in this section a sort of functions,
termed p(S)-continuous ones, and the concept of p(S)-subclosedness of the graphs of
functions, with the tacit airn of characterizing p-closedness of topological spaces. We start
with the definition of p(S)-continuity followed by sorne of its equivalent forrnulations and
certain relevant properties.
DEFINITION 3.1 : A function f : X ~ Y is called p( fl-continuous if for each filterbase 5 on
X, f(ad~ s; p(S)-adf(~, where as usual, adSídenotes the adherence of §, i.e., ad$= n{clF :
FE$}.
THEOREM 3.2 : A function f : X ~ Y is called p(S)-continuous iff for each x E X and each
preopen set W containing f(x), there is an open nbd U ofx such that f(U) s; pclW.
PROOF : Let the given condition hold and let Síbe a filterbase on X. If ad5 and W is a
preopen set containing f(x}, there is an open nbd U ofx such that f(U) s; pclW and Un f;t cj>,
for all F E ~ So, pclW n f(F) ;t cj>, for every F E $. This shows that f(x) E p(S)-adf(~ and f
becornes p(S)-continuous.
Conversely, Jet for sorne x E X and sorne preopen set V containing f(x), f(U.) r:r. pclV, for
every open nbd u. ofx. Then, §= { U.n [X\ t 1(pclV)] : u. is an open nbd ofx} is a filter­base
on X with x asan adherent point. But f(x) ¡e p(S)-adf(~ so that f(ad~ r:r. p(S)-adf(.'?.'}
THEOREM 3.3 : For a function f : X ~ Y the following are equivalent :
(a) f is p(S )-continuous.
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(b) For each x E X and each filterbase 5on X with 5~ x, the filterbase f(~ ~ f(x).
(e) For each X E X and each filterbase 5on X with 5~ X, f(x) E p(8)-adf(~ .
(d) For each x E X and every net (x..) in X with x.. ~ x, f{x..)~ f(x) .
PROOF : (a)=> (b) : For any preopen set V containing f{x), there exists, by Theorern 3.2, an
open nbd U of x such that f(U) <;;;; pclV, and then F <;;;; U, for sorne F E 5. Hence, f(F) <;;;; pclV.
(b) =>(e) : Obvious.
(e) => (a) : If f is not p(8)-continuous at sorne point x of X, there exists sorne preopen set V
containing t{x) such that [X\ t 1(pc1V)] n U*<!>. for each open nbd U of x. Then 5= {[X\
t 1(pclV)] n U : U is an open nbd ofx} is a filterbase on X such that 5~ x, but f(x) ii p(8)-adf(~ .
(a) ~ (d) : First suppose that f is p(8)-continuous, and x.. is a net such that Xa. ~ x. Consider
any preopen set V containing f(x). Then by Theorern 3.2, there exists sorne open nbd V of x
such that f{U) <;;;; pclV. Now, U being an open nbd ofx, there exists sorne 13 such that x.. E U,
for ali a.~ 13. Consequently, for ali a.~ 13. f(x..) E pclV, i.e., f(x..)~ f(x).
Conversely, let f be not p(8)-continuous at sorne x E X. Then for any preopen set V containing
t{x) and any open nbd U ofx, f{U) q:_ pc!V. Now, we define '$' on X as follows :
U $ V if and only if U <;;;; V, for any two open sets U, V in X.
Choose xu E U such that f(xv) ii pc!V. Then {xv}u is a net in X, which converge!' to x, but
{ f(Xu) }u does not pre-8-convergences to f(x).
THEOREM 3.4 : Iff : X ~ Y is a p(0)-continuous function and Y is Hausdorff, then the graph
G(f) off is closed in X x Y.
PROOF : Let (x, y) E X x Y\ G(f), then y * f(x). By Hausdorffuess of Y, there exist open nbds
U, V of y and f(x) respectively in Y such that U n V = <!> and hence U n pc!V = <j>. By p(0)­continuity
off, there exists an open set W in X containing x such that f{W) <;;;; pc!V. Then W x U is
an open nbd of(x, y) in X x Y such that (W x U) n G(f), and hence G(f) is closed X x Y.
THEOREM 3.5 : Suppose f : X ~ Y is a function and g : X ~ X x Y is the graph fnction off,
given by g(x) = (x, f(x)), for x E X. If gis p(0)-continuous, then so is f
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PROOF : Let x E X and V be any preopen set in Y containing f{x). It is easy to see that
whenever U E PO(X) and V E PO(Y). then U x V E PO(X xY). Thus X x V is a preopen set
in X x Y containing g(x). By p(0)-continuity of g, there is an open nbd U of x such that g(U) i;;;
pcl(X x V). We can show that pcl(X x V) i;;; X x pclV, and thus we have f{U) i;;; pclV, proving
the p(0)-continuity off
It is the turn of the notion of p(0)-subclosedness of graphs of arbitrary functions
between topological spaces, which will now be introduced and characterized.
DEFINITION 3.6 : A function f: X ~Y is said to have p(~-subclosed graph if for each x e X
and each filterbase $on X\ {x}with $~ x, p(0)-adf{~i;;;{f{x)}.
The proof of the following theorern, giving an equivalent description of p(0)­subclosedness
of graphs in terrns of nets, is quite clear.
THEOREM 3.7: A function f : X~ Y has a p(0)-subclosed graph ifffor each x E X and each
net (Xo.) in X\ {x}with Xo. ~ x, p(0)-adf{x..) i;;;{f{x)}.
THEOREM 3.8 : A function f : X~ Y has a p(0)-subclosed graph ifffor each (x, y) E X x Y\
G(f), there exists an open nbd U of x in X and sorne preopen set V containing y such that
[(U \{x}) x pclV) 11 G(f) = cj>, where G(f), as usual, denotes the graph off
PROOF : First, suppose that f : X ~ Y has a p(0)-subclosed graph, and (x, y) E X x Y \ G(f).
Then, y ~ t{ x). Consider r¡, * = {U \ { x} : U is an open nbd of x}. lf it is a filterbase, then r¡. * ~ x
and hence y ~ p(0)-adf{r¡. *). So, there are U E r¡. * and a preopen set V containing y such that
pclV 11 t{U) = cj>. Then Uo =U u{x} is a nbd ofx and V is a preopen set containing y such that
[(Uo \{x}) x pclV] 11 G(f) = cj>. lfr¡,* is nota filterbase, then U= {x} for sorne open nbd U ofx,
and the rest is obvious.
Conversely, suppose ~is a filterbase in X \{x} converging to x E X and the given condition
holds. Suppose y ~ f{x), i.e., (x, y) E X x Y\ G(f). Hence by the given condition, there are an
open nbd U ofx in X anda preopen set V containing y such that [(U \{x}) x pclV] 11 G(f) = cj>, or
equivalently, t{U \{x}) 11 pclV = cj>. Since, $ ~ x, it follows that F i;;; U\ {x} for sorne F E $.
Hence, f{F) 11 pclV = cj>. Then, y~ p(0)-adf{~. Consequently, fhas a p(0)-subclosed graph.
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The concepts which we are pondering upon so far in this section, will now be used to
obtain a few characterizing conditions for p-closedness.
THEOREM 3.9 : A space (Y, t) is p-closed iff for any space X, every function f : X~ Y
with p(0)-subclosed graph is p(0)-continuous.
PROOF : First, we suppose that Y is p(0}-closed and sup,pose ·f : X ~ Y is a function with p(0)­subclosed
graph, X being an arbitrary topological space. Let $be a filterbase on X and y E f{ adl:Ji¡} Then
there is an x E ad5 such that y= f\x). Let '8 ={Un F \ {x} : FE 5and U E r¡(x)}, where r¡(x)
denotes the systern of ali open nbds of x in X.
First, suppose that Wis nota filterbase. Then for sorne U1 E r¡(x) and sorne F1 E 5, U1 n F1 = {x}.
We assert that x E F, for each F E 5. If not, then for sorne F2 E $, x ¡¿: F2, Choose F3 E 5, such
that F3 s; F1 n F2. Then, (U, n f3) \ {x} s; (U1 n F1) \ {x}= 4> i.e., U, n f3 = {x} (as U1 n F3 ~ $),
i.e., x E F3 s; F1 n F2 and hence x E F2, a contradiction. Thus, f\x) E t{F), for every F E 5. Hence,
y E p(0)-adf{~.
Next, let W be a filterbase on X \ { x} . Clearly W converges to x in X. Since f has a p(0)-sub-closed
graph, f{W) has at rnost one pre-0-adherent point, viz. f{x) . Since Y is p-closed, it then follows by
virtue ofTheorem 2.13, that p(0)-adf{W) = {f{x)}. Thus, {y} = p(0)-adf{<§) s; p(0)-adf{~.
Hence, in any case, f{ad~ s; p(0)-adf{~ and consequently fis p(0)-continuous.
Conversely, to prove (Y, t) to be p-closed under the stated condition, it is to be shown, in
view ofTheorern 2.13, that every filterbase on Y has a pre-0-adherent point. Ifpossible, let there
exist a filterbase 5 on Y such that p(0 )-adf{ ~) = 4> . Let us choose and fue sorne Yo E Y Consider the
collection t* ={As; Y : Yo E Y\A}v{ As; Y : Yo E AandFs; Afor sorne FE 5}. Clearly, t* is a
topology on Y. Now, we consider the identity function f : (Y, t*) ~(Y, t). We show that fhas a p(0)­subclosed
graph. For this let y E Y and ta be a filterbase on Y\ {y} such that ta~ y in (Y, t*). Then,
by definition of't*, y= y0. For otherwise, {y} is a t* -open nbd ofy and hence the filterbase ta on Y \{y}
cannot converge to y. Also, we have 5 s; :IJ. In fact, for a gjven F E 5, {yo} v F is a t* -open ofyo and
hence contains sorne member B E ta (as$~ yo) and then B s; F. Now, in (Y, t), p(0)-adf{$) =
p(0)-ad.SW s; p(0)-ad5 (as 5 s; $") = $. Thus f has a p(0)-subclosed graph, and consequently by
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hypothesis, fis p(0)-continuous. But Yo E ad;jrin (Y, t*) whereas f{y0) = y0 ~ p(0)-adf{~ (= tj>) in (Y,
t). This contradicts the p(0)-continuity off The contradiction proves that (Y, t) is p-closed.
TH EOREM 3.1 O : Let X be p-closed. Then for ali spaces Y and Z, and ali functions f : Y ~ X and
g : Z ~X with p(0)-subclosed graphs, the set li(f, g) = {(y, z) E Y x Z : f{y) = g(z)} is closed
in YxZ.
PROOF: Let (a, b) be a limit of li(f, g). Then there is a filterbase §on li(f, g) \{(a, b)}
converging to (a, b). Ifpi, p2 denote the projection maps from Y x Z to Y and Z respectively,
we have f{p1(F)) = g(p2(F)), for ali F E $.
First, suppose that there is sorne F1 E 5 such that p1(F 1)= {a}. We obtain a filterbase /J." on li(f, g) \
{ (a, b)} by replacing only those elements F of 5 which contain b as the second co-ordinate of at
least one element, by an F* E 5 where F* s;;;; F 1 n F. Clearly, /J." converges to (a, b ). Then p2(/J.")is
a filterbase on Z \ {b} converging to b (since projection maps are continuous). Since g has a
p(0)-subclosed graph, p(0)-ad(g(p2(/J.") s;;;; {g(b)}. In this case a E p1(F) for ali F E 5 so that f{a)
E p(0)-adf{p 1 (~) = p(0)-ad(g(p2(~ s;;;; p(0)-ad(g(p2(/J.") (since /J." s;;;; ~ s;;;; {g(b)}. Thus, (a, b) E
li(f, g). In case p2(F) = {b} for sorne F E §.. we proceed similarly as above.
Finally, suppose that p1(F) :t:{ a} and PiCF) :t:{b} for ali F E $. We replace each F E $ by the
subset F* obtained by deleting from F ali those elements with first coordinate a, and thereb)'. obtain
a filterbase /J." on li(f, g) \{(a, b)} for every F E /J.", and p1(/J.") is a filterbase on Y\ {a}
converging to a. In view of the above, we can suppose without loss of generality that {b }:t:
PiCF) for ali F E /J." . Now for each F* E /J.", we consider the subset F* * of F* by deleting
from F* the elements with second coordinate b. Then /J."* = { F** s;;;; F* : F* E /J." }is a
filterbase on Li(f, g) \{(a, b)} converging to (a, b). Clearly, f{p1(F)) = g(Pi(F)) for every F E
/J."* and p1(/J."*) and Pi(/J."*) are filterbases on Y \ {a} and Z \ {b} respectively converging to
a and b respectively. Since f has p(0)-subclosed graphs, we have, p(8)-adf{p1($"*)) s;;;; { f{a)} .
By Theorem 3.9, g is p(8)-continuous and hence by Theorem 3.3 we have, g(b) E p(8)­adg(
p2(/J."*)) = p(8)-adf{p1(.9i"'*) s;;;; {f{a)} so that (a, b) E li(f, g) .
COROLLARY 3.11 : Let a space Y be p-closed. Then for any space X and any function f : X ~
Y having p(8)-subclosed graphs, the set li(f) = { (xi, x2) E X x X : f{x,) = f{x2)} is closed in X x X.
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THEOREM 3.12 : A space Y is p-closed iffor any space X and any function f: X~ Y with
p(0)-subclosed graph, the set ~(f) = { ( X1, x2) E X x X : f{x1) = f(x2)} is closed in X x X.
PROOF: Let the space Y be not p-closed so that there is a net S = {Sa : a E (D, ~)} in Y
((D, ~) being a directed set), which has no pre-9-adherent point. We choose two distinct
points y1 and y2 in Y and assurne without any loss of generality that S is a net in Y \ {y1, y2}.
Considera space (X, T), where X = Y, and T = {A k .X : A n {y1, y2}= <p} u{ A k X : A n
{y1, Y2} *- <p and Tak A for sorne a E D}, where Ta= {Sp : 13EDand13 >a}.
Consider the rnap f : (X, T) ~ Y given by f(x) = x, ifx *- Y1, Y2
ifx =y.
ifx = Y2
In order to prove that f has a p(S)-subclosed graph, Jet R = {Rp : 13 E Di}be a net in X\ {x}
converging to x (E X). Then x =Y• or Y2 (otherwise {x} is a T-nbd ofx and hence R cannot
converge to x). For definiteness, Jet x = Y•· Suppose, if possible f{R) pre-0-adheres at sorne
point y E Y. Since the net f{S) = S has no pre-0-adherent point in Y, there exists a preopen set
U in Y containing y andan ao E D such that for ali a> ao (a E D), Sa \í!: pclU, i.e., Ta k Y\
pclU. Now, {yi} u Ta is a T-open nbd ofy1 in X. Then {Rp : 13 E D1and13 ~130} k {yi} u Ta
(since R ~Y• in X), for sorne 130 E D1. Since, Tp k X\ {y1, Y2} no Rp can be Y2 for 13 ~ 130.
Thus, { f{Rp) : 13 <". 130} = { Rp : 13 <". 130} k T ª k Y \ pclU so that f{R) cannot pre-0-adhere at y,
a contradiction. Thus, f has a p(S)-subclosed graph. Now, clearly (y1, y2) \í!: ~(f). To arrive at a
contradiction it suffices to show that (y1, Y2) E cl~(f). In fact, Jet U be an open nbd of (y1, y2)
in X x X. By definition ofT, there exista, 13 E D such that (Ta u {y1}) x (Tp u {y2}) k U. lf
y E D such that y~ a, 13, then (S1, S1) E Un ~(f) proving that (yi, Y2) E cl~(f).
Cornbining Corollary 3.11 and Theorern 3.12 we get,
THEOREM 3.13 : A topological space Y is p-closed iff for any space X and any function
f: X ~ Y with p(0)-subclosed graph, the set ~(f) = { ( xi, x2) E X x X : f{x1) = f{x2)} is closed
inXxX.
Again, frorn Theorerns 3.10 and 3.12, it follows that
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THEOREM 3.14 : A topological space Y is p-closed iff for any topological spaces X and Z
and ali functions f : X~ Y and g : Z ~Y with p(8)-subclosed graphs, the set { (x. z) E X x Z
: f(x) = g(z)} is closed in X x Z.
THEOREM 3.15: A space X is p-closed ifffor any space Y and any functions f, g : Y~ X with
p(8)-subclosed graphs, the set ó*(f, g) ={y E Y : f(y) = g(y)} is a closed subset ofY.
PROOF: Let X be p-closed. Putting Y= Z in Theorem 3.10, we see that for any given space Y
and any functions f, g : Y ~ X with p(8)-subclosed graphs, the set ó(f, g) = { (y1, y2)E Y x Y : f(y1)
= g(y2)} is closed in Y x Y Now, clearly ó*(f, g) = p1[ó(f, g) n óv], where óv is the diagonal in
Y x Y and p1 : Y x Y~ Y is the first projection rnap Since, p1 b is a horneornorphism, ó *(f, g) is
closed in Y
Conversely, suppose X is not p-closed. Then for sorne filterbase § on X, p(8)-ad5= <j>. Select
any two distinct points a, b E X and put Y = X. Let, T, = {A ~ X : b E Y \ A or F ~ A for
sorne F E $}. Then (Y, T 1) is a topological space.
Consider the maps f, g: (Y, T1) ~X given by f(x) = x, for x E Y and
g(x) = X, for X E y\ {b}
=a, for x = b.
We show that each f and g has a p(8)-subclosed graph. Let y E Y and <fJ be any filterbase on Y\
{y}such that <fJ ~y in Y Ify ~ b, then {y} being a T1-open nbd ofy, <fJ cannot converge to y.
Hence, y= b. Now, for each F E $,Fu {b} E T,, so that p(8)-ad(<fJ) = p(8)-ad<fJ ~ p(8)-ad.91=
<j> ~ { f(y)}. Hence, f has a p(8)-subclosed graph. Similarly, g has a p(8)-subclosed graph. Now,
ó*(f, g) =Y\ {b}, and we see that for any open nbd V ofb in Y, there is an F E $ such that F ~
V and V n ó *(f, g) ;;;? F n ó *(f, g) ~ 4> (as F ~ {b}, for each F E ~. Hence ó *(f, g) is not closed
in Y
§ 4. STRONGL Y p(8)-CLOSED GRAPH ANO p-CLOSEDNESS
DEFINITION 4.1 : A function f : X ~ Y is said to have strongly p(~-closed graph G(f) if
whenever a net x.. ~ x in X and f(x.,) ~y in Y, it follows that y= f(x).
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THEOREM 4.2 : A function f : X~ Y has a strongly p(8)-closed graph G(f) iffwhenever a
filterbase .~~ x in X and f{~ ~y in Y, it follows that y= f{x).
PROOF : Straightforward and left.
LEMMA 4.3 : A function f : X ~ Y has a strongly p(8)-closed graph G(f) iff for each point
(x, y)~ G(f), there existan open nbd U ofx in X anda preopen set V in Y containing y such that
(U x pclV) 11 G(f) = ~.
PROOF : Let the given condition hold for a function f : X ~ Y, and let xu be a net in X such
that Xa ~ x anti f{x..)~ y in Y. It then follows that for each open nbd V of x and each
preopen set W in Y containing y, (V x pclW) 11 G(f) *- ~. Hence, (x, y) E G(t) and so, y = f{x).
Hence, fhas a strongly p(8)-closed graph.
Conversely, suppose that a function f : X ~ Y does not satisfy the stated condition of the
theorem. Then for sorne (a, b) E X x Y\ G(f), we have (U x pclV) 11 G(t) *- ~. for every open
nbd U ofa and each preopen set V containing b. Suppose, ~ = {Ua : a E 11 and Ua is an open
nbd of a} and .9; = { pclV p : 13 E h and V p is a preopen set containing b}, and put ~ = { ~ . .., P>
= (Ua X pclVp) 11 G(f) : (a, 13) E 1, X h}. Then $= {F(a, Pl: (a, 13) E 1, X h} where F(a, Pl =
{ x E Ua : (x, f{x)) E W<a. p¡}, is a filterbase on X such that $'converges to the point a in X,
f{~~ b and f{a) * b. Thus in view of Theorem 4.2, f does not have a strongly p(8)­closed
graph.
In view ofTheorems 3.8 and 3.9 and Lemma 4.3, it now follows that
THEOREM 4.4: IfY is a p-closed space, then every function ffrom any space X to Y with a
strongly p(8)-closed graph is p(8)-continuous.
DEFINITION 4.5 : A space X is called pre-Hausdorff if corresponding to any two distinct
points x, y ofX, there exist disjoint preopen sets U, V such that x E U, y E V.
THEOREM 4.6: A pre-Hausdorffspace Y is p-closed ifeach function ffrom any topological
space X to Y with a strongly p(8)-closed graph is p(8)-continuous
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PROOF : Let, Xo e Y and Jet (Xa)aeo be a net in Y\ {Xo} with no pre-0-adherent point in Y\
{xo} . Let X= {Xa : a e D}v{Xo}, and Jet T be the topology on X generated by { {Xa}: a e O}
v {Tμ v {Xo} : μe O} as basic open sets, where Tμ = {Xa: a<::μ, a e O} . Let f : X---+ Y be
the identity map and let (x, y) e X x Y\ G(f). Ifx ~ Xo. then {x} is open in X. Choose, by pre­Hausdorffuess
of Y, a preopen set V in Y, containing y with x ~ pclV. Then clearJy ( { x} x pclV)
11 G(f) = <j>. Ifx = Xo, then y~ Xo, so there is a preopen set V containing y satisfying Xo ~ pclV,
as Y is pre-Hausdorff Again y~ Xo, so y is not a pre--0-adherent point of (x..) in Y. Hence,
there is aμ e D such that Tμ 11 pclV = <j>. Thus, Tμ v {Xo} is an open nbd ofx in X and V a
preopen set in Y containing y such that [(Tμ v {Xo}) x pcJV] 11 G(t) = <j>. Hence, the graph off
is strongly p(0)-closed and by hypothesis, f is p(0)-continuous. So for any preopen set V
containing Xo in Y, there exists μ e D satisfying T μ ~ pclV. Thus, x.. ~ Xo in Y. Hence, Y
is p-closed.
That the assumption of pre-Hausdorffuess on Y is essential can be observed from the
following resuJt :
THEOREM 4. 7 : If a surjection f : X ---+ Y has strongly p(0)-closed graph, then Y is pre­Hausdorff
PROOF : Let y and z be any two distinct points of Y. Then since f is onto, there is x e X
such that f(x) =y. Hence, (x, z) i: G(t). Since fhas a strongly p(0)-closed graph, there existan
open nbd U of x and a preopen set V containing z such that f(U) n pclV = <j>. Put W = Y \
pclV, then W is a preopen set containing y and W n V='· Hence Y is pre-Hausdorff.
Combining Theorems 4.4 and 4.6 we obtain :
THEOREM 4.8 : A pre-Hausdorff space Y is p-closed iff each function from any space X
into Y with strongly p(0)-closed graph is p(0)-continuous.
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