Download o PAIRWISE LINDELOF SPACES

Re.v,u,ta. Coiomb.£ana de. Ma.te.mmc.M
VoL
XVII
(1983),
yJlfgJ.J.
31 -
58
o PAIRWISE LINDELOF SPACES
by
Ali
A.
FORA
and
Hasan
L
HDEIB
A B ST RA CT. In this paper we define p:rirwise Lindelof
spaces and study their properties and their relations
with other topological spaces. We also study certain
conditions by which a bitopological space will reduce to
a single topology.
Several examples are discussed
and
many well known theorems are generalized concerning Lindelof spaces.
RESUMEN.
En este articulo se definen espacios p-Lindelof y se estudian sus propiedades y relaciones
con
otros tipos de espacios topologicos.
Tambien se estudian
ciertas condiciones bajo las cuales un espacio bitopologico (con dos topologias) se reduce a uno con una sola topologla.
Se discuten varios ejemplos y se generalizan varios teoremas sobre espacios d~ Lindelof.
1.
I nt roduce
pological
ion.
Kelly
[6] introduced
space, i.e. a triple
T , T are two topologies
1
2
the notion of a bito-
(X,T1,T2) where X is a set
on X, he also defined pairwise
and
Haus-
37
dorff, pairwise regular,
generalizations
pairwise
normal spaces, and obtained
of several standard results
theorem.
such as Urysohn's
Lemma and the Tietze
extension
have since considered
the problem of defining compactness
such spaces: see Kim [7], Fletcher,
san [1].
Cooke and Reilly
Several
authors
for
Hoyle and Patty [4], and Bir-
[2] have discussed
the relations
be-
tween these definitions.
In this paper we give a definition
bitopological
of pairwise
Lindelof
spaces and derive some related results.
We will use p- , s- to denote painw~e
ively, e.g. p- compact,
and semi-compact
s- compact
and
~emi-, respect-
stand for pairwise
compact
respectively.
The L.-closure,
L.-interior
l
of a set A will be denoted by
l
Cl.A and IntiA respectively.
The product topology of L1 and L2
l
will be denoted by L1xT2.
Let R, I, N denote the set of all real numbers,
val [0,1], and the natural
Ll' L
numbers respectively.
U6ual, ~o~ountable,
denote the ~enete,
r
kigth-~y
topologies
the inter-
Let Ld, L , L ,
u
c
lent-~y
and
on R (or I).
,
2.
Pairwise
Linde15f
Spaces.
Let us recall known defini-
tions which are used in the sequel.
2.1
[4J. A cover ~of
is called L 1L 2 -0 pen if
'U..
the bitopological
eLl
U L 2'
If,
space (X,L1,L2)
in addition,
contains
u..
at least one non-empty
member of L1 and at least one non-empty
member of L2, it is_called p-open.
2.2
[4]. A bitopological
space
lS
called
p-~ompa~ if
every p-open cover of the space has a finite subcover.
38
2.3
[3]. A bitopological
ry T T -open
1 2
2.4
space is calle
s-~ompact if eve-
cover of the space has a finite subcover.
space (X.T1.T2) is called T1 ~ompact with nehpect ~o T2 if for each T1-open cover of X there is
a finite T2-open subcover.
[1]. A bitopological
2.5 [1]. A bitopological space (X. T1.T2) is called
B-eompact if it is T1 compact with respect to T2 and T2 compact
with respecto
to T .
1
If we replace
definitions
the word "finite" by the word "countable"
2.2, 2.3 and 2.4, then we obtain the definition
p-Lindelofi, s-Lindelofi,
~pect ~o
and (X,T1,T2) ~
T2, respectively.
2.6
(X,T)is
Lindelofi with
of
ne-
space (X,T1,T2) is called B-Lindelon
with respecto to T2 and T2 Lindeloff with
A bitopological
if it is T1 Lindelof
respecto to T1.
It
T1
in
clear that (X,T1,T2) is s-Lindelof
if and only if
Lindelof
where T is the least-upper-bound
topology
lS
of T1 and T2. It is also clear that if (X,T1,T2) is B-Lindeloff
then each (X,T.) must be a Lindelof
space for i
1,2.
=
l
2.7
Wh~n we say that a bitopological
a particular
topological
property,
space (X,T1,T ) has
2
without referring specially
to T1 or T2, we shall then mean that both T1 and T have the
2
property; for instance, (X,T1,T2) is said to be Hausdoff if
both (X,T1) and (X,T2) are Hausdorff.
THEOREM 2.8.
The
bdopofogiea1. !.lpaee r x, T l'
T
2)
i!.l e-Lin39
delCi6 -<-6 and only -<-6 d ~ Li.-nde1.o6 and p-Li.-ndelCi6·
P~oo6. Necessity
follows
inmediately
from the observation
that any p-open,L -open
or L2-open cover of (X,Ll,L2) is L 1L 2-0l
not
pen. Conversely, if a L1L2-open cover of (X,Ll,L2) is
p-open, then it is Ll-open or L2-open.
EXAMPLE 2.9.
The bitopological
space
(IR,Ld,Lc) is p-Lin-
delof but is not s-Lindelof.
EXAMPLE 2.10.
defined
Consider
the two topologies
L l' L 2 on R
by the basis
=
{(-ro,a):a > O}
U
{{x}:x
> O} , and
~2 = {(a, ro):a < O}
U
{{x}:x
< O} .
~l
Then
(IR,L ,L ) is p-Lindelof but is not Lindelof.
It is also
l 2
clear that (R,L ,L2) is not B-Lindelof, for the Ll-open
cover
l
. {(-ro,l)} U {{x}:x ~ l} of R has no countable L2-open subcover.
2.11 [8J. A bitopological space (X,Ll,L2) is called
p-Qountably 'Qompact if every countably p-open cover of X pas a
finite
subcover.
2.12
A bitopological
space
able Qompact if every countably
(X,Ll,L2)
L1L2-open
is called
S-Qount-
cover of X has a fin-
ite subcover.
2.13 A bitopological space (X,Ll,L2) is called
ably Qompact with ~~pect to L2 if for each countably
cover of X there
2.14
is a finite
A bitopological
L2-open
space
2
countably
compact
The following
40
Ll-open
subcover.
B-QOunt-
(X,Ll,L2)
is called
compact
with respect
albtj Qompact if it is L1 countably
and L
l-Qount-
L
with respecto
fact is obvious:
to Ll.
to L2
THEOREM 2.15.
s B)-c.ountably
p(Jte-6p.
n
lo
compact and p(Jte-6p.
s,B)-countably
(ii) EveJty p(Jte-6p.
s ,B)-compact
EvVty p(Jte-6p.
(i)
-6pace J.A
s B)-Lindelon.
compact p(Jte-6p.
s,B)-Linde-
J.A p (Jte-6P. s, B) - compact.
-6pace
EXAMPLE
p-Lindelof
2.16.
The bitopological
space (R,Ld,LC)
space which is neither p-countably
is a
compact nor p-com-
pact.
EXAMPLE
2.17.
Let L s denotes the Sorgenfrey
topology
on R. Then the bitopological
space (R,L ,L ) is s-Lindelof
u s
but
is not B-Lindelof, because the L -open cover {[-n,n):n EN}
s
of R has no L -open countable subcover.
It is also clear that
u
the space (R,L ,L ) is neither
u s
EXAMPLE
(N,Ld,Ld)
2.18.
s-countably
compact nor s-compact.
It is clear that the bitopological
is B-Lindelof
but is neither B-countably
space
compact nor
B-compact.
THEOREM 2.19.
-6pace
In (x , L l'
Let <e
J.A a hVteciUaJty
Lindelo n
n.
-then d i-6 s- Lindelo
PftO06.
T 2)
{U : a E i\} U {VS:S e; I"] be a L L -open
a
1 2
cover of X, where Ua e: L 1 for each a e: I\. and V S e: L 2 for each
BE
U
=
=
U{Ua a
e:
r} is
L
countable
I\.} is L1-Lindelof, there exists
a
set 1\.1c I\. such that U = U{ua:a e: 1\.1}'
Similarly,
since V =
U{VS:S
f.
Since
E
2-Lindelof, there exists a countable
U{VS:S e: f1}.,It is clear
that
set f1 c f such that V =
{U :a € I\. } U {VS:S e: f } is a countable
a
-6pace
1
1
COROLLARY
2.20.
J.A s - Undelo
6.
EvVty
subcover
sesiond countable
of ~ for X.
bdopologic.al
41
EXAMPLE
Let X = RXI and < be the lexicographical
2.21.
order on X. Let
=
~1
{[x,y):x
< y ; x,y E X}
and
~2
=
{(x,y]:x < y ; x,y EX}.
Let T ,T be the topologies on X which generated by the basis ~1
1 2
and ~2' respectively.
Then (X,T1,T2) is a Lindelof space which
is not p-Linde16f,
because
the p-open cover
{ [ ( a , x ) , (1 , x )) , ( ( a , x ) , O,x)] :x
of X has no countable
neither
s-Lindelof
Lindelof
=
subcover.
It is clear that (X,T1,T2)
nor B-Lindelof.
is
2.22.
but it is s-Lindelof.
EXAMPLE
~2
Id
Let X and T1 be the same as in example
Then the bitopological space (X,T1,T2) is not hereditary
EXAMPLE
2.21.
E
2.23.
R, '81 = {X,{x}:x E X-{O}}
and
Let T1,T2 be the topologies on X which are
Let X
{X,{x}:x E X-{l}}.
=
by the bases ~1 and '82' respectively. Then (X,T1,T2)
is B-Linde16f, for any T1-open cover of X or any T2-open.cover
generated
of X must contain X as a member.
However,
(X,T1,T2)
is not p-Lin-
de16f, for the p-open cover {{x}:x E X} of X has no countable
subcover ,
We may summarize
some of the above examples
and theorems
by the diagram on tre nextpage (T stands for theorem while E stands
for example).
2.24
[7J.
If T is a topology
subset of X then the
topology
42
on X and A is a non-empty
adjoint topology (denoted by T(A)) is the
on X defined by T(A) = {¢,X} U {AUB: BET}.
NO
E2.23
NO
E2.17
o:;:----------------.~
s-Lindelof
B-Lindelof
YE
o
T2 8 E
2.21
I
NO
E2.10
Lindelof
----------------~
p-Lindelof
2.25
A family
every member of
2.26
7 of
nonvoid subsets of X is T1T2-closed
is T1-closed or T2-closed.
Y
[8]. A family
1of
nonvoid T1- or T2-closed
sets
if 1contains members F1 and F2 such that F1
in X is p-cto~ed
is a T1-closed proper subset of X and F2 is a T2-closed
subset of X.
2.27 [6].
weakly
A set U in a topological
ope~ if for any p
ing p such
E
space (X,T)
proper
lS
called
U there exists an open set V contain-
that V~U is a countable set.
ly arosed if X-F is weakly open.
then p is called a weak-~ntenio~
weakly open set V containing
weak-interior
if
A set F is called weak-
If A is a subset of X and p EX,
po~
of A if there exists
p such that V cA.
The set of
a
all
points of a set A is denoted by WInt A.
It is clear that WInt A is the largest weakly open set
contained
in A. It is also clear that WInt A
A is weakly open, and Int B cWInt
LEMMA 2.28.
be a weakly
crosed
Let (X,
p~op~
T ,T )
1 2
=
A if and only if
B for any set B eX.
be a p-L~~de1.o6 -6pac.e.a~d c
.6ub~et i~ r x, T 1)' The~ c ~
T 2- L~~-
de1.°n·
43
P~o06.
Let C be a nonempty
of X and let ~ = {va:a
L1-weakly
closed proper subset
A}
be a L2-open cover for C. For each
x E X-C there exists a L -open set H(x) such that x e: H(x) and
1
H(x) n C is a countable set. Since C ~ ~ and C f. X,
the
n
{V :a
a
E
EA} U {H(x):x E X-C}- is a p-open cover for the p-Lindelof
space X.
A1 c A and a countE A } U {H(x ),H(x2),··}
1
1
Thus, there exists a countable
set
able set {x ,x , ...} c X-C such that {va:a
1 2
is a countable cover for X. Since H(x. )n C is countable
l
00
for all
00
ie: lN, the set Cn(UH(x.))
is countable, say CO(UH(x.))
=
i=1
l
i=1
l
{Y1'Y2, ... L Since Yi e: C, there exists ai e: A such that Yi e:
' It is clear now that {va:a e: A1} U {Va.:i e:
ai
l
able subcover for C. Hence C is L2-Lindelof.
V
Since every closed
lowing corollary
W} is a count-
set is weakly closed, we have the fol-
to Lemma 2.28.
COROLLARY
A
2.29.
6
delCi 6 -6pac.e £6 L j- LLndelCi
L i-uo-6ed
(i f. j; i,j
Using a similar technique
COROLLARY 2.30.
~pac.e i6 Lj-c.ompact
It is important
j; i,j
=
-6ub-6eX. 06 a p-L{.n-
1,2).
as above, we obtain the following;;;
A Li-uo-6ed
(i f.
p~op~
=
pftOP~
-6ub-6eX. 06 a p-c.ompact~~~
1,2).
to note that the word "proper"
in Lemma
2.28 can not be removed.
not Ld-Lindelof
For example, R is L -closed but R is
c
in example 2.16.
We now obtain
delof spaces.
44
four alternative
characterizations
of p-Lin-
OoUowing Me. e.qLUVa£e.ru:.:
p-Lindel0n.
(a) (X,11,12) ~
(b)
p-clo~e.d
Ev~y
oamity with
ty ha6 no nempty iru:.~
p~op~-
1, the topology T2(V) ~ Linand oo~ each non-empy ~et v in T2, the topology
delao,
11(V) ~
(e)
int~eetion
eetio n,
~et V in
Fo~ each non-empty
(c)
the. countable.
1
Lindeloo·
Each 11-weak1.y clo~e'd p~op~
and each
1
2
~u.b~et
-we.ak1.y clo~ ed p~op~
P~006. The
The equivalence
00
X
.6u.b~et:
fact that (a) is equivalent
cs T2-Lindelein,
0 -6 X ~
T 1-
to (b) is obvious.
of (a), (c) and (d) can be obtained
ogous way to the proof of [2,Theorem
plies (e) is due to Lemma 2.28.
Lindelo -6.
in an anal-
2]. The fact that (a) im-
The fact that (e) implies
(e)
is obvious.
An easy characterization
found in the following
THEOREM 2.32.
delan
io
and only
io
of s-Lindelof
spaces can be
theorem.
A bitopologiea£
e.v~y
able inteM e.etio n pM p~y
~pace (X, 1 l'
T 1 1 2-clo~ed
T 2)
oamity w..{;ththe
s- Lin-
~
count-
hiu. nonempty iru:.~ eetio n.
,
,
THEOREM 2.33.
be
B-
Lindela
O.
EXAMPLE
Let (X,T1,12) be B-compaet and (Y,T1,12)
Then (Xxy,11x1~,12X1;) ~ B-Lindela6.
2.34.
Let 1f denote
the cofinite
topology
on
R. Then (R,lf'ld) is p-compact.
Howeve·r, the space
2
(R ,lfX1f' ldX1d) is not even p-Lindelof, for the p-open cover
{R x(lR- {O})} U {( x, 0) :x
2.35
[9].
e:::
IR} of
jR2
A space (X,11,12)
has no countable
subcover.
is said to be p-Ha~do~oi
45
if, for any
distinct
points x and y, there
hood U of x and a l2-neighbourhood
We observe
a l1-neighbour-
V of y such that·U
that if (X,ll,l2)
and l2 are T1-topologies.
p-Hausdorff spaces.
is
is p-Hausdorff,
The following
theorem
n V = ~.
then both l1
characterizes
The. ooUow-lng
ptwpeJLtie.J.> aJte. e.qtU.val.e.n-t:
(a) The. b,[topofog-lQal. ~paQe. (X,l1,l2) ~ p-Ha~do~oO'
(b) Fo~ e.a~ x g X,
THEOREM 2.36.
{x}
=
n {ci
U :U
1 a. a.
a. Et:.
lS
a l2 neighbourhood
of x}
and
{x} =
(c)
n {Cl
a. ez I:
The. diagonal.
2
U:U
a
a
is a l1 neighbourhood
D = {(x,x):x
x} ~
g
of x}.
a elo~e.d ~ub~e.t
,[n
e.ach
On
the. ptwduct topofog-le.J.> (XXX,l1Xl2) and (XXX,l2Xl1)'
P~oo6. (a) implies (b). Let x E X and y E X such that y i- x ,
By (a) there exists a l1-open
that y
e:
V l'
X
set V1 and a l2-open set V2 such
E V 2 and V 1 n V 2 = 4>. This implies that y e: C11 ~2'
This proves the first part of (b). The proof of the second part
of (b) is similar to the one we just proved.
(b) implies
(c).
Let (x,y)
e:
XXX-D.
Then x,y
e:
X and x
y. By the second part of (b), there exists al1-open
set U1
containing x such that y e: X-C12U1·
Let U2 = X-C12U1·
Then U2
is a l2-open set and it is easy to check that (x,y) e: U1xU2 c
XXX-D.
Hence D is a closed set in the topological
(XxX, l1Xl2)'
space
In a similar way we can prove that D is l2xl1-
closed subset of XxX.
(c) implies
EXXX-D.
46
(a). Let x,y
e:
Since D is a l{l2-closed
X such that x
i- y. Then (x,y)
set, there exists a l1-open
i-
set U
and a T2-open
is clear now that x
1
Recall
section
Then
set U2 such that (x,y) E U1xU2 e XXX-D.
e:: U , Y e::U
U = <jJ •
2 and U1
1
2
that a space (X,T) in which every countable
P~006.
T. ]
Let A be a T.-Lindelof
eros ed (i ~ j;
subset and x e::X-A.
l
(\
{Cl.U:U
o,e::
I::.
l 0,
is a T. neighbourhood
0,
a T.-open
cover of the T.-Lindelof
a countable
set A.
l
l
set
1::.
1
el::.such that {X - CliUo,:a
{X - Cl.U
l
0,
I::.}
:0. e::
Thus there exists
e:: 1::.
} is a cover
1
n
]
]
Using the same technique
COROLLARY 2.38.
T. -
of x}
U X-Cl.Uo," Let U =
U.
Then U is a
ae::1::.1 l
o,EI::.1a
set,contains x and U eX-A.
Hence A is T.-closed.
for A, i.e. A e
f1.Y
By
]
(i ~ j; i,j = 1,2). Since A e X-{x}, therefore
T.-open
i , j = 1, 2 ) .
2.36 we have
{x} =
lS
inter-
is called a p-~paQe.
.s uiis et: M
T.-Lindelo6
l
Theorem
n
of open sets is open,
ev~y
It
Qompac.t
-6 ub~
l
e;t M
Le;t
as above we obtain
(X,T1,T2)
T. - ctM ed
(i
~
the following.
be p-Ha.u6doJz.66. Then
j;
i, j
]
=
eve-
1,2).
[6] .
In a space (X,T ,T2), T1 --L6 ~cu:.d to be Jz.egui.M
1
wUh Jz.e-6
pea to T2 if, for each point x in X and each T1-closed
set P such that xl- P, there are a T1-open set U and a T2-open
2.39
set V such that x e::U, P e V and U
n
V =
<jJ
(X,T1,T2) is p-Jz.egui.a.Jz.
if T1 is regular
T2 and vice versa.
2.40
[9J. In a bitopological
space
with respect
to
(X,T1,T2), we say that
47
It is interesting
to note that if L1 is regular with respect to L2 and L2 is coupled to L1, then L1 CL2.
Thus, if
(X,L1,L2) is p-regular and Li is coupled to Lj (i I j, i,j =
1,2) then L1 = L2 and the resulting single topology is regular.
,
r
It is also interesting
to note that if L1 is coupled to L2 and
r
r,
L1 is regular with respect
to L2 then (X,L1) is regular.
[6] .
2.41
A space (X,L1,L2) is said to be p-no!U11l:Lt if,
given a L -closed set C and a L2-closed set F such that en F = ¢,
1
there are a L1-open set G and a L2-open set V such that F c G,
Cc
V and V
n
G = <jJ.
THEOREM 2.42.
~pac.e
p-Jtegu1.aJt,
p-no!U11l:Lt.
EveJty
p- Li.Yl.deio6
bdopologic.a1.
(X, L1,L2) ~
PJtoon. Let A be a nonempty L1-closed set and B be a nonempty L -closed set with An B = <jJ.
Since (X,L1,L2) is p-regular
2
for each a E A, there exist a L2-open set Ga and a L1-closed set
F
with a L G c F c X-B. Also, for each b E B, there exist a
a
a
a
L1-open se t- eband a L2-closed set Mb with b e: Cb c Mb eX-A.
Let
= {Cb:b E B} U {X-B} and ~ = {Ga:a EA} U {X-A}. Since
e
e
and ~ are p-open covers
exist countable
the p-e Li.ndeLof space X, there
for
subcollections
{C1,C2, ...} of ~ and {G1,G2, ...}
00
of g such that A c; UG. and B c Ve .. Let V1 = C1 and, for
i=1 l
l=1 l
n-1
each positive integer n > 1, let Vn = e - UF..
For each
n
n
i=1 l
00
and
UM.. Let V = Uv
positive integer n, let H = G
n
n
i=1 l
n=1 n
co
l' H liE L 2 ' A cHand
B c V. Furthermore,
H = UH . Then VEL
n=1 n
x E H n V, then x E H n Vn for some m and n, and so
m
m
n-1
x e: (G M.) n (C - U F .). Considering separately the
m
i=1 l
n
i=1 l
cases m > nand m ~ n yields a contradiction and so H n V = <jJ.
00
U
Thus (X,L1,L2) is p-normal.
48
•
Let X be a fixed non empty set, and
'BX = {( X,T ,T '):T and T I are topologies
~ on 'BX by:
Define the partial ordering
Then we have the following
Let
THEOREM 2.43.
on X} .
theorem.
f = {(
X, T ,T') E <EX:(X,T ,T ')
b,
a
p.LLnde1.on P-.6pac.eJ and jf = {( X, T ,T') e: 'Ex: r x, T ,T') b, a p-HalL6don66 p-.6pac.e}. 16 (X,T1,T2) E in 1/, then (X,T1',T2) b, a rniiU.ma1. e1.eme.nt
06 !f and a
mlx.Una1. e1.emerz,t
06 i..
Pnoon. Suppose (X,T~,T~) Elf such that (X,T~,T~) ~ (X,T1,T2).
Therefore
T~ c: T1 and T~ c: T 2' Let G
T1-closed
proper
Corollary
e:
T 1-H}. Then X·-G is
a
subset of X. Since (X,T1,T2) is p-Lindelof,
by
.-.
2.29, X-G is T2-Lindelof.
But T; C:T2" Therefore X-G
is T~-Lindelof.
Since (X,T~,T~)
lary 2.37, X-G is T~-closed.
is p-Hausdorff
P-space,
Hence G e::T~. Consequently
by corolT1 = T~.
In a similar way we can show that T2 = T~.
Now, let (X,T~,T~) e::£ such that (X,T1,T2) ~ (X,T~,T;).
,
,
,
Then TiC: T 1 and T 2 c: T 2' Let U e:: T i-a}. Then X-U is a T i-closed
proper
subset of X. Since (X,T~,T~)
is p-Lindelof,
by Corollary
2.29, X-U is T~-Lindelof.
But T2 c: T;. Therefore X-U is T2-Lindeis p-Hausdorff P-space, by corollary 2.37,
lof. Since (X,T1,T2)
X-U is T i-closed. Hence U
,
E:
T l' Consequently
I
=
lar way we can show that T2
Using the same technique
T 1 = T t ' In a simi-
T2.
as above we obtain the following
theorem.
Let
THEOREM 2.44.
pad},
and
(X ,T 1 'T 2) E
.re = {(X,T,T
,
I
) E'13X:(X,T,T ) b, p-c.omJv6 p-HalL6don66}.
16
Jf = {(X,T,T') E<EX:(X,T,T')
e n Jf, then (X,T 1 'T 2) b, a mLvUmal. e.1emerz,t a 6 jf and
a maximal. e1.emerz,t
an
~.
2.45
49
(p-op~n, p-cto~~d,
iff f:(X,L1) +
(Y,0 ) and f:(X,L ) + (Y,02) are continuous(open, closed, homeo1
2
morphism, respectively).
0 VLtornp
Let f:(X,L1,L2)
2.46.
THEOREM
uo lL6
p-hom~omo~phUm,respectively)
+
(Y,01,02)
be. a P-c.on..t.in-
.
Ifi
(X,L1,L2) ~
16
f ~
p-Lindelofi (s-Lindelo6, B-Lindelo6, ~~p~c.tivefy), (Y,01,02) ~ p-Lindelo6 (s-Lindefo6, B-Lindelo6, ~~p~c.tivefy) .
(b) 16 (X,L1,L2) ~ p-c.ompact (s-c.ompac.t, B-c.ompac.t, ~~p~c.tiv~fy), then (Y,01,02) ~ p-c.ompac.t (s-c.ompac.t, B-c.ompac.t, ~~pec.tivefy) .
(a)
one-to-one., (Y, 01'02) ~ p-HaU6do~fi6 P-~pac.e. and
(X,L1 ,L2) ~ p-Lindefo6,
then. f u a home.omo~pYU~m.
(d) Ifi f ~ one.-to-one., (Y,01,02) ~ p-HalL6do~6fi and (X,L1,L2)
~ p-c.ompac.t, the.n f ~ a homeomo~p~m.
PMo6. (a) let e= {va:a e:: Ll} U {ua:a e:: Ll} be a p -open cover
(c)
of V such that
"« e:: 01
and Ua e:: 02 (a e::Ll).
Then
{f-1 (V ):0'. e::.6. } U {-1
f (U ):0'. e:: Ll} is a p-open cover of X because
a
a
f is p-continuous and onto.
exists a countable
-1
{f (Va) :ae::Ll1
}
U
Since (X,L1,L2)
Ll such that
is p-Lindelof,
there
set Ll1 C
f (Ua) :ae::Ll1 } is a cover for X. Thus
{-1
Ll } U {ua:a e:: Ll1} is a countable subcover of <e for X.
1
The remaining parts of the statement (a) are similarly proved.
{va:a
e:
(b) The proof is similar to that in (a).
(c) It suffices
to show that f is p-closed.
proper subset of X.
Let A be aLl-closed
Then, by Corollary
2.29, A is L2-Lindelof.
f:(X,L2) + (Y,02) is continuous.
Hence f(A) is O2-Lindelof because
By Corollary 2.37, f(A) is 01-closed.
Similarly,
it can be shown
that the image of every L2-closed subset of X is a O2-closed
subset of Y. Hence f is p-closed.
(d) The proof is similar
50
to that in (c).
3. Conditions
reduced
under
to
a
which
single
space
is
topdldgy.
3.1. Let
THEOREM
a bitopological
be. a HaU6do!l.66 p-Undelo6
(X,L ,T )
1 2
The.n T1 = L2.
P!l.oofi. Let G E L1-{~}. Then X-G is a T -closed proper sub1
set of X. By Corollary 2.29, X-G is T2-Lindelof. By Corollary
p-~pace..
2.37 we have:
"every Lindelof
(X,T) is closed".
Thus X-G is
2-closed, i.e. G
we have T2 CL1·
Consequently
L1 C L2· Similarly
THEOREM
~pace..
subset of a Hausdorff
Let
3.2.
P-space
T . Hence
2
T1 = L .
2
L
E
be. a compact p-HaU6do!l.66
(X,T1,T2)
The.n T1 = T2.
P!l.oo6. Let G E T1.
compact
space (X,T1).
2.38, X-G is L2-closed,
we have T2
C
Then X-G is a T1-closed subset of the
Therefore X-G is T1-compact. By Corollary
E
T2.
Hence T1
C
T . Similarly
2
Thus T1 = T2.
T1·
LE~1A 3. 3.
Let
(X, L l'
F be. a T1-we.aQty eio~e.d
T 1- Unde..f..o6 .
P!l.oo6. Let q
G containing
i.e. G
E
~et
L 2)
be. a p- Undelo
~uch that
WInt2(X-F).
fi ~ pace.
and .f..e.:t
1 ~. The.n F ~
WInt2(X-F)
Then there exists
q such that G n F is a countable
a T2-open set
set. Let ~ =
6} be a T1-open cover for F. For each x E X-F there
exists a T1-open set H(x) containi?g x such that H(x) n F is a
{Ca:a
E
countable
set.
Since {Ca:a
cover for the p-Lindelof
61
C
6 and {x x ' ...}
1 2
{ca:a E6 }
1
C
E
6}
U {G} U {H(x):x
E
x-rI
is p-open
space X, there exist two countable
sets
X-F such that
U {G} U {H(X ), H(x2), ...
} is a cover for X.
1
00
Since H( x.)
l
n F is countable,
the set F
n (U H( x ,»
i=1
is countable,
l
51
Let ui e 6 such that y.l e::Cu.l
say {Y1'Y2' ... }.
is a countable
{Cu:u e: 61} U {CU{ie: N}
L1-Lindelof.
We can use the same technique
lowing
theorem
"empty"
(We replace
in the proof
THEOREM
Let
3.4.
Ld-closed
is not Ld-Lindelof.
1 ~" is a necessary
.opae.e, and let
by the word
Let
1 ~. Then F ~
Int2(X-F)
set which
3.6.
the fol-
(X, L l' L 2) be a p-e.ompact .opae.eand let
In the p-Lindelof
THEOREM
p.
"countable"
3.5.
rrWInt2(X-F)
i.e. F is
3.3).
F be a L1-c..fo.oed.oet .oue.h that
pact.
EXAMPLE
e,
of
as above to conclude
the word
of Lemma
subcover
(i e:IN). Then
(IR,Ld,Lc)' 1R is a
This shows that
condition
in Lemma
3.3.
be. a p-Hau.odo~66 p-L~de.fo6
(X,L1,L2)
u be a L i-weaU!!
space
L1-e.om-
open
.0
et:
e.on.t.cUru.ng a 6-ixed po-<.nt
Then
( a) thvr.e.
ewtoo L 1-open
sets
that
p e::nc. e Fe
i=1 l
(b) ecthe»: p e::C12(X-U),
Ci(i
e: N)
and a Ll-c..fo.oed.oet F .oue.h
U; and;
on, thvr.e. ewt
and a L 1-c..fo.oed set: F .oue.h that
J,2-open.ow
QGi
p e::
Gi
Ci e::lN)
e FeU.
P~oo6.
exists
(a) Since p e: U and U is L1-weakly open set, there
a Li-open set A such that p e: A and A-U is a countable set.
For each x e: X-U there
exist
G(x) such that x e: G(x),
An
B(x).
D(x)-U
a L1-open set B(x) and a L2-open set
p e: B(x) and B(x) n G(x)
Let D(x)
= ~.
Then Pe: D(x), D(x) e::Ll,
52
{G(x1),G(x2),
set.
=
<f>,
and
X-U is a L1-weakly closed proper
subset of the p-Lindelof space X, by Lemma 2.28, X-U is L2-Lindelof. Therefore the L2-open cover {G(x):x e: X-U} has a countable
subcover
is a countable
D( x ) n G(x)
=
Since
... } . Since
D(xi)-U
(i e: N)
is a count-
00
UD( x .)-U lS countable,
i=1
l
e: IN).
Then Ci (ie:~)
able set, the set
Ci = D(xi)-{yi} (i
be~ause (X, T 1) is a T1-space.
((1C .) n G( x .) = ¢ for all j
i=1 loo
J
e: F.
i.e. (lC.
i=1
F e: U.
Hence p
nC.
e:
i=l
Therefore
e:
C12(X-U),
p
e:
T1-Lindelof.
00
(n
i=1
J
00
n
i=1
Hence
e: IN.
Since {G(x.):j
l
(b) If P
Let F =
say {Yl ,Y , ... }. Let
2
is a T -open set
1
X-G( x. ). Then
e: lfr}
l
C .) n
l
00
(U G( x . »
i=l
is a cover for X-U,
then
e: F e: U.
then we are done.
Suppose p ~ C1 (X-U).
2
~ ¢. By Lemma 3.3, X-U lS
Int2U,
i.e. Int2(U)
For each x e: X-U there exist a T -open
1
set G(x) such that p e: G(x), x e: C(x)
set C(X)
and
The T1-open cover {C(x):x e: X-U} has a count{C(x1),C(X2), ... } . Let G. = G(x.) and F =
able subcover
00
l
00
(lx-C(x.).
i=1
¢
l
and a T2-open
C(x)
G(x) = ~.
n
=
J
p
Then
e:
l
(lG.
i=1
l
Using the same technique
we get the following
l
e: Fe: U.
as in the proof of Theorem
3.6,
theorem.
Let (X,T1,T2)
be. a p-HalL6doJtoo p-c.ompact
U be. a T1-ope.n ~et c.ontaining
a oixe.d point p.
THEOREM 3.7.
~pac.e. and, let
The.n
(a)
the.Jte. e.~t
p
(b)
e:
T
C e: F e: u,
eithe.Jt p
T 1-
a
e:
C
and a
T 2-uMe.d
se:
F
such. that
and
on: the.Jte. exist:
C12(X-U)
UM e.d ~ et:
se:
-ope.n
1
F /.)uc.h
that
p
e: G
e:
a
F C::
2-ope.n
T
sei:
G
and
a
u.
p-HalL6doJt6o p-c.ompac.t ~pac.e. ~
gutaJt (and he.nc.e.~ by The.oJte.m 2.42, ~ p-noJtmal).
COROLLARY
3.8.
p-Jte.-
A
PJtooO' Use Theorem 3.7 (a).
COROLLARY
3.9.
A p-HalL6doJtoo p-Linde.ioo
p-Jte.guiaJt (and he.nc.e., by The.oJte.m 2.42~ ~
p-~pac.e.
p-noJtmal).
~
53
P~oo6.
Use Theorem
COROLLARY
~pac.e..
I6
3.7 (a).
Let (x;r1,L2)
be.
all U E L1-{x},
3.10.
1 ¢ 6o~
Int2U
a p-Ha£L6do~66 p-c.ompae.t
then L1
C
L2.
P~o6. Let U E L1-{X}. Then X-U is a L1-closed set with
Int2U 1 ~ . By Theorem 3.4 X-U is L1-compact. Hence, by Corollary 2.38, X-U is L2-closed,
be. a p-Ha£L6do~6 6 p-c.ompae.t
6o~ all U EL1-{x},
and Int1V 1 ¢ 6o~ all
COROLLARY
~pac.e..
v
I6
i.e. U E L2.
Let (X,
3.11.
L l' L 2)
1¢
The.n. L1 = L2.
Int2U
2-{xL
« L
P~oll'
Use corollary
It is interesting
ed a theorem
3.10.
to note that Cooke and Reilly
[2, Theorem
4J for B-compact,
spaces but did not get any analogous
For this reason,
Theorem
Corollary
p-~pac.e..
6o~ all
result for p-compact
is an extension
I6
1 ~ 6o~ all
Int2U
the.n. L1
2
1~.
Let (X,L1,L2)
3.12.
V E L -{X},
P~o6.
Int2U
=
[2,
U
The.n. th~e
CUte
-
F Midl
.
L2.
Let (X, L l'
Unde.lo 6 ~ pac.e..
( X, L l)a
54
of the result
Let U EL1-{xL
Then X-U is a L1-closed set with
By Lemma 3.3 X-U is Ll-Lindel~f.
Hence by Corollary
THEOREM 3. 13.
~et
spaces.
be. a p-Ha£L6do~66 p-Unde.e: L -{x}, and Int V 1- t
1
1
2.37, X-U is a L2-closed set, i.e. U EL2.
ly we can prove T2 c T1·
Thus T1 = T2.
p e:: u.
and bicompact
4] .
COROLLARY
106
3.11
s-compact
[2J obtain-
00
that
P
E:
(\
Let u
L
i=l
l
C
C
L2. Similar-
be. a p-Ha£L6do~6 6 ~ pac.e. an.d
be. aLl -we.ak£y 0pe.n. ~ et: and
L 2)
-OPe.n. ~W
2
G.
Thus L1
FeU.
G
i
Ci
E
IN)
and a
1-c.lMed
T
Pnno6. For each x E X-U there exist a T2-open set G(x)
and a T1-open set H(x) such that x E H(x). p E G(x) and
G(x) n H(x) = ~.
Lindelof
T1-open
Since X-U is a 11-weakly
space (X,T1), therefore X-U is 11-Lindelof. Thus the
cover {H(x):x ~ X-U}oohas a countable subcover
{H(x1),H(x2),
...
L Let F = (lX-H(x.).
Take G. = G( x . ).
FeU.
closed set in the
l
l
Using a similar
l=l
ool
Then p E (1 G.
i=l l
technique
Then F is T -closed
1
c FeU.
and
as above we can prove the fol-
lowing theorem.
Let (X,T1,T2) be a p-HalL6doftnn .6pac.e and
(X,1 ) a c.ompact .6pac.e. Let u be a T -open. sei: and p E u. Then
1
1
tit eJte a.Jte a T 2 - apen .6et: G an.d all - c.1.0.6 ed .6e;t F .6uc.h that
THEOREM 3.14.
pEG
c
FeU.
COROLLARY
p-.6pac.e,
then
Pftoo6.
3. 15.
1 = T2.
Use Theorem
In
T 2)
-i.J.> a Lin.de1..o6 p-HalL6doft6
n
1
3.13.
Since every B-Lindelof
have the following
COROLLARY
and UtheJt
(x,« l'
space is Lindelof,
we
corollary.
6
As a corollary
16
(X, 1 ,1 ) -i.J.> p-Ha.lL6doft66 P-.6pac.e
1 2
Oft s- Unde1..o 6, then. T 1 = 12,
3. 16.
B- Lindelo
to Theorem
sult (see [2, Theorem
COROLLARY
(s-Lindel~f)
re-
4J).
3. 17.
B-c.ompact Oft s-c.ompact,
3.14 we have the following
16
then
(x , T 1' T 2) -i.J.> p-HalL6doft6 nand
1
1
WheJt
= T2.
55
4•
Con c 1 us ion.
generalizations
As we
noted
of well-known
tension of some theorems
, our results
classical
in this paper are
theorems
as well as ex-
in the literature.
Naturally,
any result stated in terms of T1 and T2 has a
'dual' in terms of T2 and T1. The definitions of separation and
covering
properties
dorff and p-Lindelof,
and covering
of two topologies
1 and T2, such as p-Hausof course reduce to the usual separation
properties
T
of one topology
T , such as Hausdorff
1
when we take T1 = T2; and the theorems quoted above then yield
as corollaries the classical results of which they are generalizations.
As an example of theorems
results
are theorems
Theorem
2.15, 2.33, 2.36, 2.42, 2.43, 2.44 and 2.46.
2.8 is an analogue
2.3 (a,c,d) is an analogue
that Corollary
2.18J.
to [2, Theorem
to [2, Theorem
3.11 is an extension
3.7 (a) implies the results
Theorem
which yield well known classical
2].
We notice also
of [2, Theorem 4J. Theorem
in [4, Theorem
12 and 13J and [7,
It is also clear that Corollary
ogue to [7, Theorem
1J while Theorem
2.9J and Corollary
2.30 is an anal-
2.38 is an analogue
[7, Lemma 2.11J. It is clear too that Theorems 2.8, 2.42
Corollary
2.20 imply the result
to
and
in [6, Lemma 3.2].
REFERENCES
dans les espaces bitopologiques", An..
15 (1969)
317-328.
Cooke, I.E. and Reilly, I.L., "On bitopological compactness",
J. London Math. SOQ. (2), 9 (1975) 518-522.
Datta, M.C.-, "Pr-ojec t Ive bitopological spaces", J. AtL6:tJtai..
Math. $OQ. 13 (1972) 327-334.
Fletcher, P., Hoyle III,H.B. and Patty, C.W., "The comparison
of topologies", Vuke. Math. J. 36 (1969) 325-331.
Birsan,
T., "Compacite
~t.Univ. Ia6i, ~.I.a., MatematiQa,
56
[5]
Hdeib, H.Z., "Contribution to the t heor-y of [n,m]-compact
paracompact and normal spaces", Ph.D. Thesis (1979)
SUNY at Buffalo.
Kelly, J.e., "Bitopological spaces", Pltoc.. Londol1 Ma;th. Soc..,
13 (1963) 71-89.
Kim, Y.W., "Pairwise compactness", Pubf. Math. Ve.blte.C.e.I1,
15
(1968) 87-90.
Pahk, D.H. and Choi, B.D., "Notes on pairwise compactness",
Kyungpook Math. J. 11 (1971) 45-52.
Weston, J.D., "On the comparison of topologies", J. Londol1
Math. Soc.. 32 (1957) 342-354.
~':~':
Mathe.matiC6 Ve.paJttme.11t
IGi.ng Saud Un.-iVV1.J.lfty,
Abha, SAUVI ARABIA.
Ve.paJttme.YLt06 MathematiC6
YaJrmO uk Un.-iVV1.J.lfty
IJtbid, JORVANIA.
(Recibido en abril de 1981).
57