31
Internat. J. Math. & Math. Sci.
(1990) 31-38
VOL. 13 NO.
ON NORMAL LATTICES AND WALLMAN SPACES
GEORGE M. EID
Department of Mathematics
John Jay College of Criminal Justice
The City University of New York
445 West 59th Street
New York, NY 10019, U.S.A.
(Received March 28, 1988 and in revised form June 20, 1988)
ABSTRACT.
Let X be an abstract set and
lattice of
a
subsets of X.
The notion
Also, the general
of R being mildly normal or slightly normal Is investigated.
not necessarily
Wallman space with an alternate topology is investigated, and for
disjunctive, an analogue of the Wallman space is constructed.
KEY WORDS AND PHRASES. Normal lattice, 0-1 valued measures, Wallman space, disjunctive
lattice, almost compact, almost countably compact.
1980 AMS SUBJECT CLASSIFICATION CODES. 28A60, 28A52.
I.
INTRODUCTION.
In the first part of this paper, we consider lattices which satisfy conditions
We
weaker than normality; more precisely mildly normal and slightly normal lattices.
examples
give
of
such
lattices,
and
then
investigate
the
preservation
of
these
properties under lattice extension and restriction.
Next, we investigate spaces which are related
First,
instead
of
considering
the
customary
to
topology
the
general Wallman space.
on
the
Wallman
space,
we
introduce another topology and show how topological properties reflect strongly to the
underlying lattice.
Then we consider the case of a lattice which is not necessarily
This work generalizes
disjunctive and construct an associated Wallman type space.
that of Liu (see Section 5).
We adhere to standard lattice terminology that can be found, for example, in [I],
[2], [3], [4|, [5].
However, in section 2, we summarize the principal lattice
concepts and notations that will be used throughout the paper for the convenience of
the reader.
We then precede to the consideration of mildly normal and slightly normal
lattices in section 3, and then to analogues of the general Wallman space in sections
4 and 5.
We flnally note that most of the results hold equally well for abstract
lattices.
2.
DEFINITIONS AND NOTATIONS.
.
a) Let X be an abstract set and fl a lattice of subsets of X.
We shall always
assume, without loss of generality for our purposes, that
X E
The set whose
general element L’ is the complement of an element L of
is d.,ed by Q ’.
,
’
32
G.M. EID
said to be complement generated if f, for every L of
{Ln}n=
such that L
in
v of X,
x
B’
=I,l=1
.
y of X,
is
said
L then there exists an
for every x E X and every L E
if x
LI,’ L CL
x
LIfL 2
and
.
L OL2’
{L};
of
L
i
1
be
two
lattices
abbreviation
L
exists
IE
(
2 ,
.
ill’ 2 of
2) iff,
s.s.q
such
if f,
LlC LI, L2C L
L
X.
of
fr and L flL^IL
every
fll
.
QI
E
if,
.
E
X and L
is
,
and
regular if f,
fl such that
.
if
L L2’
is
then for a countable subcollecton
fl is compact tff, for
subsets
2
A’, y e B’
every x
for
L and L O L
such that
2
L2C L
that
.2 such that x
is T
L.
L and y
L then there exists L I, L
2
is normal Iff, for any L|, L2g
then for some finite subcol[ecttou (L
any
E
if x
A, if
Li= ’"
A,B
disjunctive
such that x
E i,
L
;
lindelof Iff, for every L
(La
L!
then there exists
for any two elements
such that
there exlst,
to
,
there exists a squence
is separating if
there exists an element I,
for any two elements x
and A’
L’.
n
of
everv
and every
I
is
,
.
L
{L};
said
is
Lcze
to
e A, if
Next, consider
or
semi-separate
L2E
if
said
to
Llfl L2=
for
0 then there
for
if
separate
L
L I,
and
such that L
there exist L I, L
2
2
2
We denote by I :, qet whose general element is the intersecti,,
arbitrary subsets of fl. b) Let A h, ,.,y algebra of subsets of X. A measure on A is
R such that
is bounded and finitely
from A
defined to be a function,
If x
X,
additive. The algebra of subsets of X generated by fl is denoted by A(.).
any L 2,
L! 0
then
The
if L flL
L2c
is the measure concentrated at x so u..’.\)
x
element is a measure on A()
set whose
general
if x.A
where A
if x#A
Note that, since
is denoted M().
[0
every element of M(fl) is equal to the difference of nonnegatlve elements of M(),,
without loss of generality, we may work excl..’i voly with nonnegative elements of M(fl).
eM(fl),
Let
is ft- regular if for any A e
A(); (A)
sup{(L);LCA, L
E
fl}.
The set whose general element is an element of M()which is -regular is denoted
by
MR(fl).
An element
E
M(fl) is o-smooth on fl,
if
Lne
fl, n
The set whose general element is an element of M(fl)
1,2,... and
Ln
then
o-smooth
1,2...
o-smooth on A(fl) if An A(fl), n
o
0. e set whose general element Is an element of M(fl) which
and A + t’ iCA
n
smooth on A(fl) is denoted by M(fl). Note that if
MR(Q) then
are the subsets of the corresponding M’s
M (fl).l() IR(fl) I () l(fl)
For U E M(fl), the support
consisting of the non-trlvlal zero-one valued measures.
then
L Is replete iff, whenever
of
e
S(,)
u(X)}.
E fl, u(L)
A’premeasure on fl Is defined to
S() #
a function
from fl to {0,I} such that
() 0, (A) < (B) for every A CB where A,B and If (A)
then
(B)
I. H() denotes the set of all premeasures on G. c)
(AB)
an immediate
consequence of Zorn’s Lemma, we have, for every B
I(), there exists an element
(L
0.
on fl is denoted by M ().
We say that
w:[ t, i,
is
l(fl)
,,
.
I()
IR() such that U v on
GI’G2 of subsets of X, if GIC
that
E IR() such
vIA(G
v
Moreover,
is normal iff, for
,
or simply
(B
v(fl)).
then for every
and
such
I(),
that
<
I(G)
E
Also, for any two lattices
IR(G1 )’ there exists a
a v is
unique
if
v2(G)where
G1
separates
I,2
IR(
.
then
33
ON NORMAL LATTICES AND WALLMAN SPACES
I
R is regular iff, for any i,2 e l(fl);
u2"
result
xe IR(fl)
Iff fl is
leads
disjunctive
B2(fl)
I
the
to
us
then S(
S(2).
I)
Topology
Wallman
The
which
is
for L e fl} as a
{ e IR(fl); (L)
obtained by taking the totality of all W(L)
base for the closed sets on IR(fl). For a disjunctive fl, IR(fl) with the W(fl) of closed
sets is a compact T space and will be T Iff fl is normal and is called the general
2
Wallman space associated with X and fl. Also, for a disjunctive fl and A,B
(W(A))’,
W(A) is a lattice with respect to union and intersection. Moreover,W(A’)
W(A(R)) -A(W(fl)), W(A) -W(B) iff A B and W(A)c W(B) iff AcB. Now, we note that,
E M(R), there exists a
if
is disjunctive so is W(R), and in addition to each
is
A(R) such that the map
E M(W(fl)) defined by (A) -(W(A)) for all A
one-to-one and onto; moreover
3.
E
MR(R)
Iff
E
MR(W(R)).
ON NORMAL LATTICES.
In this section, we elaborate on the notion of a normal lattice, investigate
lattices which satisfy weaker conditions, and discuss their interrelations under the
will
and
Throughout this section
and restriction properties.
extension
denote
iff
,
R’RI’
lattices of subsets of the set X.
LICLUL
Note that, for
R; i=1,2,3; O is normal
Lle
AIUBI; AL, BICL;; AI,BIE
then L
I(R) and G
{L’s R’.LCL’, B()
Let R be normal,
then, G is a prime R’-filter.
and if
PROOF. Clearly, G is an fl’- filter since
G, (LIOL2)
and by the
L
G, then LC
G. In
THEOREM 3.1.
,
L UL
If^ I’vL
then^L^ with addition,^
L
+ (2
(L)
.
I)
B(LI)
LIUL 2 G
filter.
G is a prime
G.
or
Thus,
Lv
L
’
Then,
C
separates
THEOREM 3.2. Suppose
and
then either (L
either
RI
i
normal.
LIE G;LICL
,
^nrmallty
or (L
I
is
2)
of
and so
normal
iff
is
is
Suppose fl is normal then by the separation. It [s clear that
Suppose R2 is normal. Let
normal.
(ii)
I(I )’
I’ 2 (RI); I’ 2 (RI )"
there exists an
Suppose
to
Extend i,2, to
e IR(R2) and
I().
PROOF. (i)
I’ %2
L2efl2;
1(L2)
(f12)
v2
II(L 2)
O, then
I(L)
1. By the separation, there exists
ll(L 1)
z
1()
1
1, but
but z(L
1)
then
and similarly z
ll(LI)
(f12 ).
and
1’ LI
R’
there
sL1DL2’
LIDL2"
exists
,
which is a contradiction.
Thus, by the normality of
L2 ; L2CL
2
Consequently,
Therefore,
1
then
and so O is normal.
DEFINITION 3.1.
R is said to be mildly normal, if for all
v(R).
I (), there exists
o
IR(R);
DEFINITION 3.2. R is said to be almost countably compact, if for all
t
Iz(n’), v : I o(n).
s()
DEFINITION 3.3. R is said to be prime replete, if for all
THEOREM 3.3. If R is regular and prime replete, then R is mildly normal.
PROOF. If R is not mildly normal, then there exists
IR(R) and
I
fl;
Vl(L 1) v2(L2)
2
Vl(fl), < v2(fl). Then, there exists L 1,
L
is
S()
since
regular,
But
0.
and Vl(L 2)
S(Vl)C and
v2(L1)
so
S()=
and
@ which is a contradiction
S()
S(v2)CL 2. Then, S()C(LlfL 2)
is prime replete. Thus, R must be mildly normal.
since
a unique v
lo(n),
.
Io(R),
L2
L1OL
,+
(;.M.
34
THEOREM 3.4. II
PROOF. If l a
.
S()CL[OL2)
), but
h
L miLdLy
L feB,fLat ;nd I.lndolof, Chea
miLdLy ,,maL. then b the proof ,f mor
r,
reover,
S(v l)
(p)
:;(v) wJh
,t
,
for air t uhtch ts a contrdXclon stnct,
,()
I(L)
u c I ().
is normal.
THeOReM 3.5. If l te alos co,,nbty ,.ompac snd nttdly normal, hen
v IR(’)
PROOF. uppoe 2 it almo co,**tl-ably compact, and u *: I(). en
(), then v (
(). If g I R(O) ,nd u
a so v < u(); v
obvou.
I (fl). SInCe B Is mildly ,,-rmal the ret of the poot
Then, f12 Is mtdLy nori If
p,rCes
THeOrEM 3.6. Suppose sljc
und
’l
’
on
v
,
2
01
is mildly normal.
PROOF.
Suppose
Xm tldly u,rml.
sl
U < Vl(t2),
<
--co,in,ably unded, tf given
gnl
t p
Io(),
t.
mildly nomo L.
DEFINITION 3.4.
f2
t slid Lo
2
21
(),
, n,
In ehe ne heorem, e will eee hen a partial converse of uhe eheorem 3.6
THEOREM 3,7. Suppos
Suppose l2
PROOF
u
v
(Vl(fll),
a,d v
v2(l)
<
separates
l
Let
Bn #’
n
c
Vl,V2
l
r
s
and
is mildly nu,ttdl and
whera
2 exend uniquely to
l’ 2(L2 )’
s,
tr(rt).
1
t sl
-counubly unded, hen
eparsem
Extend
it
to
I
t.
t c
I(f12).
(1)
and
By
IR(2)’
and 2 respectively, where l’ 2
t,e
ta l-countsbly bounded, there
and
’
I
.
S mildly normal.
v2 o
v
if whenever A
paracompacc
couutably
DE?INTION 3.5. 9 I sald to be
there exlsCs B c fl;AncB and ’;
1hLly no1, t for all 9 e I (9). here
i ,ald to
ZFINtTIUN 3
S,ce
exists
2
i,
mildly normal.
IR():
unique v
I
2
,,d llenee
<
not.
c I(I’), N
Suppome tm not mltghtly tt, t-ml. Let
1(), g ()
L
(L’);Ln;
Sap
Let
(L)
where l,2e IR(); v
2’ en S(U)
S() aince fl ta regular.
(). H,,reover, S(t,)
en, tm a prmeasure, i.e.
which is a contradicAlso
o(), and aince 0 ts lind,of, S() # then S()
nurmat.
li.ghtly
tion. Thus, 2 mut
THEOM 3.8.
[f ] iI regular atd tlndelof, then fl Is slightly
PROOF.
,.
,
ttghly cormal.
PROOF.
N
X
();
i(22),
v2(n2)
her v t,
v2e IR(12),
then by
35
ON NORMAL LATTICES AND WALLMAN SPACES
iI 21
is slightly normal and so
since
I
by separatlon and hence
v2
f12
is
slightly normal.
LEMMA 3.1.
PROOF.
is slightly normal.
is complement generated, then
If
Io
Thus,
is complement generated, then
If
IR();
there exists a unique v
(fl).
fl’ )C
1R(fl),
Io( fl’
that is, if
is slightly normal.
In the nexf theorem, we will see when a partial converse of theorem 3.9 is true.
3.10. Suppose
separates f12" If
f12 is mildly normal and countably
THEOREM
I
paracompact then
is slightly normal.
fll
Io(I’);
2(I);!’ 2 IR(I). Extend to
I(2).
BnC A’n +’ AnE I since 2 is countably
Bn +’ BnE
paracompact and
I separates 2 therefore 2 is I -countably paracompact and
(Bn) (A) --(A) 0. Since E I(i), then
la(2). Now extend
to
and so
then
separation
E
2(2)bY
i(),
TI, T2 IR(2)
l,V2
I 2
is mildly normal. Thus
is slightly normal.
since
and so
I
I 2
PROOF.
Let
E
I(I )’
Consider
REMARK 3.1.
then
It is not difficult to give similar conditions (as in theorem 3.10)
to obtain the other partial converse of theorem 3.9.
ON SPACES RELATED TO THE GENERAL WALLMAN SPACE.
4.
We
X,
recall
IR()
with
from
the
for
that
2
section
topology W() of
an
lattice R of
arbitrary
the closed set, is a compact T
is disjunctive and separating then X can be embedded in
if
disjunctive, so is W(), and
IR()with
this section, we consider alternate topologies on
THEOREM 4.1.
IR()
2
is
T
Also,
space.
Moreover, if
is
is normal.
In
iff
IR().
z W(’) for a base of closed sets W(L’);L’E P
Consider
with the topology
IR().
the topology W()is T
of
subsets
Then
,
2.
I(LI
2:I’2 IR() then there exists L I,L 2 ;LIrL2
and I(L2)
O. Therefore I W(LI )’ 2 W(LI and
and 2(LI)
0, 2(L2)
and
2 E W(L2)’ I W(L2); W(LI) and W(L 2) are open sets. Thus, W(LI)rW(L 2)
is T
consequently, IR() with
2.
DEFINITION 4.1.
is said to be almost compact, if for all
E (’), S()
PROOF. If
B1
#
THEOREM 4.2.
PROOF.
^
S()
Let %
E
,
The lattice W(’) is almost compact.
E
IRW((’)’)
IR(W())
(W(L))ffi B(L).
W(L)with
then %
with W(’) is almost compact.
IR()
E
W(L)
Thus,
.
REMARK 4.1. We note (i) S(k)= S()ffi {}. (li)If
L L’, LCL’; L e
E
clear that for any L
THEOREM 4.3.
PROOF.
,
The sets of W() are clopen in the
is
-topology.
THEOREM 4.4.
IR(
(i) Suppose
Let
and so is W(’).
is
topology.
closed in the -topology.
(W(L’)’).
IR()with
compact.
I(W(’)). Then S()
IR()
is
then
is
also
,
But, W(L)
Thus, the sets of W()are closed in the
with the -topology is compact iff
is
it
disjunctive,
Since W(L) is disjunctive then by remark 4.2 (li) for any L e
W(L) =r.W(L’), W(L)CW’(L ),L
open in -topology since W(L)
PROOF.
(section 2) and so
S( ) on W() and so
E
is an algebra.
Thus, W(’) is a compact lattice
on W(’),
;
k
E
I().
G.M. EID
36
(W(L’)) iff
Let V e S(), V
IR(n) then
Now A(L’)
I. Then (W(L’))
But V e IR(L)
IR(W(fl)). Thus,
A(L).
hence B
(W(L’))
I.
iff A(L’)
I. Thus,
then v e W(L’) then v(L’)
< (fl’) and
then V- A. Thus
and hence l(W(fl))
’
[2]. (li) The converse is clear.
W(fl) Iff fl is an algebra.
PROOF. (i) Since the sets W(fl) are clopen by Theorem 4.3 then W() e 6 and so
is
Now, if 6
TW(.) then since W() is compact so is 6 and so
W()
6.
algebra by theorem 4.4. (ll) The converse is clear.
l(W(fl’))
THEOREM 4.5.
6
an
In the next theorem, we glve another equivalent condition for fl to be an algebra.
is an algebra iff W(fl’) is a disjunctive lattice in IR().
e IR(fl);
PROOF. (1) Suppose
is an algebra, then IR()- l(fl). Let
u((L’)’). L’ e and W(L’)NW((L’))’
0,
W(L’), L e ft. Then u(L’)
(L)
U
$. Thus, W(’) is disjunctive. (li) Suppose W(’) is disjunctive. Let
there exists
). For
e IR();
u e I() then there exists a
0, L)
then v 4 W(L’). Hence, by disjunctlveness
L e
,(L)
W(( U L)’). Hence,
W(L’ N L’)
W(L)’W(L)’
e W()’, L e n and
THEOREM 4.6.
,
,
LULffi X, L’OL’
and
8o
L’CL, but since |(L)
.
which is a contradiction, since
Thus,
IR()
A
W()’,
and
(L)
I(), and so
0
is an algebra.
5. ON NON-DISJUNCTIVE LATTICES.
is not necessarily disjunctive. We begin, by
We next consider the case where
introducing the notion of an -convergent measure and some related results and then
proceed to the construction of an analogue of the Wallman space.
x
X such
e I() is said to be -convergent if there exists an x
DEFINITION 5.1.
that
( )"
on ’, for all
S()
is -convergent iff
THEOREM 5.1.
e I().
is f-convergent. Then there exists x e X; Ux u() and so
(1) Suppose
on n’.
(ll)
(n’)" Moreover x
S(x )C S(,) on n’. Thus, S()
e I(). Let x e S() on ’. Then, B
Suppose S() # on n’ for
x(n’) and so
is f-convergent.
()" Thus
THEOREM 5.2. Suppose
"I 2 ()’ for all I’ 2 e I(). Then
PROOF.
" x
x
a)
If
I
is -convergent so is
b)
If
’
is regular and
PROOF.
Suppose
a)
2 () then
I
Bx
f?-convergent, then
but
I
S(I)
U2 ()
on
THEOREM
then
’
x
2
and so
’
LIL
y e
then
L2;
u(LI)
is T
2
u(L2)
result is true.
is -convergent, then
x
2
and
’
’
I
I is
Suppose
5.3.
If
I
I
is it-convergent.
UI()
for some x, but
2 () and so 2 is -convergent. b) Suppose 2 is
(’) and x e S( 2) on
2 () for some x and so 2
exists a unique x e X:
PROOF.
2
2"
is f-convergent then
xand
and since
S(BI
S(2
on
’
’. Thus,
it-convergent.
T
is
2
and
is f-convergent where
u e I(). Then,
there
(’)"
if x#y then there exists L I,
I. Now, if
I,
x(L2)
x(Ll)
but
’
x
is regular x e
LI0
L
2
which
is
a
L
x
2 e
;
U(n)and
LI,
y
and
so
contradiction
x e
B()
the
desired
ON NORMAL LATTICES AND WALLMAN SPACES
37
ts said to be weakly compact if for all
DEFiNITiON 5.2.
IR(),
is
-convergent.
.
DEFINITION 5.3.
on
is said to be almost compact if for any
THEOREM 5.4.
x
then
IR() then S(B)
p (x(’),
x
IR(’),
S()
weakly compact iff ]’ Is almost compact.
is
PROOF. (i) Let
IR( then since
such that
() and S()
on ’.
e
+:
on
’
is weakly compact, there exlsts an x E X
us,
ls almost compact. (li) Let
Is almost compact. Let x
S() on
is weakly compact.
since
(() and so
’
’
Now, note that a topological space X is absolutely closed (generalized absolutely
closed) iff the lattice
of open sets Is T2(To) and weakly compact. Next, let
be a
lattice of subsets of X and define U()
THEO 5.5. a)
b) Suppose
If
and
l
l cU(l)and 2
L
{UL
E
}.
Is weakly compact, then
2
2" en,
seml-separates
I
if
is weakly compact.
PROOF.
a)
[
is weakly compact.
Is weakly compact,
Extend
IR(I) to v IR(2). Since 2 is weakly compact, there
(I and so $I is weakly compact, b) Let v e IR().
Bx 2
x (where
en since is.s2
is the restrlction of v to A(LI). us, there
E IR([
exists an x
X such that
)"
suppose L
Now,
and x(L2)
then x g X
x (l
2
exists an x;
X
but L
L2,
2
U
and moreover
(LIe
x 92
us,
)"
)"
(
LI
L
I ; then x
x
since
L]
is on some
([)’
but
and so
LIecL 2
x (L
then
for some L
v(L2)
and so
is weakly compact.
2
t X be a topological space and e the collection of open sets.
en, by eorem 5.4, e is weakly compact Iff F e’ is almost compact.
Now consider X. Suppose 9 is non-dlsjunctlve and define I
{x: x E X} U
A
I;
I,
{ IR(): U Is not convergent} and W(A)
{
A()}. We
(A)
also
REK 5. I.
assume
is
To,
X and x
so x, y
y implles
Bx
"
#
For A, B
B iff W(A)
A(), we have: a) A
W(B), b) W(AU B)
W(A) U W(B), c) W(A O B)
W(A) W(B), d) W(A’)
(W(A))’, e) W(A())
A(W(9)).
PROOF. a) (i) If A
B, then, clearly W(A)= W(B). (li) If A B, then say
A OB’
let x e AOB’ then x(A0 B’)
I and so x(A)
I, x(B)
0 which
I;
implies that
W(A),
W(B). b), c), d) and e) are not difficult
4W(B)and W(A)
THEOR 5.6.
,
x
to show and are omitted.
Now
en,
e I() and
consider
one
can
e I(W()) to
define
Is
that
note
easily
I-I
and
be
onto
(A), A
(W(A))
from I() to
A().
I(()),
and
moreover B
IR() iff E (W()).
THEOM 5.7. W() Is weakly compact and T
PROOF. a)Let
concentrated at
S(U
I R(W()) then
x
if
x(W(A)) =0I if
implies that
x(A)
E
m I
R().
< (W()). Note that for A
x and
x
A
x
A
and
so
A
W(L) with u(W(L))
measure concentrated at
,
x
E
W(A)
o
If
Is convergent then
x
(
()
A(),
iff
x(A)
Thus
x
is the measure
W()-convergent. If B is not convergent then
A
u(L), hence U E W(L), U e I, U e S(U) and
is the
and consequently W() is weakly compact, b) Let
is
U
I’
G.M. EID
38
I,
U2
l
Ul
0.
To.
W(L’) and so W(L) Is
U() then W() separates U(W()).
THEOREM 5.8. If
PROOF. Suppose (i) (U (La) O
0" Let A
Therefore
u2(L)
I,
I(L)
such that say,
then there exists an L
^2
W(L),
U2
=(ULa )
(W(L))
e a, B
(L 8)
e
a.
cW(B). If AnB
and
I then L nL
U() then
W(L which
)and
W(L
e
e
and
for some
B- Let x e L 0L B then
0, and the desired result is now clear.
contradicts (i). us, A B
0, W(A)W(B)
Since
a
(UW(LB)
(UW(Lo))cW(A)
,
x
x
x
B)
W() is generalized absolutely
U() then I with
Now, we note that If
we
tff
u,
ensider X and
is
eIed and t absolutely closed if
X tnce ne can easily
then I, 0 is an absolute elure
and
(a)
2"
2’
obere that X
(X).
o, o.
(A)
Where
analogous construction can now be done for i
{ e I O,
Is not convergent} and
{x: x e X} U{ e
is weakly replete
Is
weakly
replete.
that
show
one
can
A
and
I,
A()
(A)
O
here
constructions
the
for any
if
IR() g is convergent. We note that
o
REK.
l(fl):
genraite the ork
()
Lut [6].
REFERENCES
I.
NOEBELING,
G.,
Grndlagen
der
analytlschen
topologle,
Sprlnger-Verlag,
Berlin
1954.
C.I.P.,
Comm.
Math.
Carollnae,
13
2.
FROLIK, Z., Prime Filters with
(1972), 533-575.
3.
BACHMAN,
4.
GRASSI,
5.
SZETO, M., Measure Repleteness and Mapping Preservations, J. Indian Math. Soc.
43(1979), 35-52.
6.
LIU, C.T., Absolutely Closed Spaces, Trans. Amer. Math. Soc. 130 (1968), 86-104.
7.
BOURBAKI, N., Elements of Mathematics, General Topology, Part I, AddisonWesley, Readlng, Mass. 131.
the
Univ.
r-smoothness, and
G. and STRATIGOS, P., Criteria for -smoothness,
tightness of Lattice Regular Measures, with Applications, Con. J. Math.
33(1981), 1498-1525.
P., On subspaces of replete and measure
Bull. 27(I), (1984), 58-64.
replete spaces.
Con. Math.