Itnat. J. Math. Math.
Vol. 4 No. 2 (1981) 289-04
289
THE ORDER TOPOLOGY FOR FUNCTION LATTICES
AND REALCOMPACTNESS
W.A. FELDMAN
Department 6f Mathematics University of
Arkansas, Fayetteville, AR 72701
and
J.F. PORTER
o
Department
Mathematics, University of
Arkansas, Fayetteville, AR 72701
(Received June 12, 1980)
ABSTRACT.
A lattice K(X,Y) of continuous functlonson space X is associated to
each compactlflcatlon Y of X.
It is shown for K(X,Y) that the order topology is
the topology of compact convergence on X if and only if X is realcompact in Y.
This result is used to provide a representation of a class of vector lattices with
the order topology as lattices of continuous functions with the topology of cornpact convergence.
This class includes every C(X) and all countably universally
complete function lattices with i.
It is shown that a choice of K(X,Y) endowed
with a natural convergence structure serves as the convergence space completion
of V with the relative uniform convergence.
KY WORDS AND PHRASES. Order topology, relative uniform convergence, realcompactness, universally complete function lattice, convergence space completion.
1980 MATH4TICS SUBJECT CLASSIFICATION CODES.
1
54A20, 54C40, 46A40
INTRODUCTION.
In this paper we will study a broad class of real function lattices which we
call "2-unlversally complete."
For this class we will show that the order top-
ology T o(also called the order bound topology and the relative uniform topology)
is the topology of compact convergence in an appropriate representation
(Theorem2).
We will show (Proposition 3) that the 2-universally complete lattices include the
290
W.A. FELDMAN AND J.F. PORTER
"C(X),
lattices
all continuous real-valued functions on
universally complete lattices containing
i.
X,
and all countably
(An example of this latter type is
discussed in Example i.)
The proof of Theorem 2 requires a construction which is studied independently
Y
X are investigated.
of
K(X,Y)
In particular, sublattlces
i.
in
C(X)
for compactifications
Theorem i states that the order topology for
is the topology of compact convergence on
in
of
X
if and only if
(Th/s concept of realcompactness was studied in
Y.
X
K(X,Y)
is realcompact
[I0].
Since the order topology is the finest locally convex topology in which
(see [5]), in
every relatively uniformly convergent net converges
consider the 2-universally complete function lattice
convergence as a convergence function lattice
associated order topology.
Vp,
3
we
V wth relative uniform
rlthout reference to its
We show (Theorem 4) that
K,Y)
endowed with a
natural convergence structure serves as the completion in the convergence space
sense of
Vp.
We remark that (assuming without loss of generality that X is realcompact)
it can be seen directly that the order topology is the topology of compact
C(X).
convergence for the lattice
This follows from
[13,
p.
124]
since every
positive linear functional is continuous with respect to the topology of compact
convergence (see [6]) and since
is barrelled
2.
CCX)
with the topology of compact convergence
(see [11]).
THE ORDER TOPOLOGY FOR
Let
Y be a compact Hausdorff space and X a dense subspace.
by F(X,Y)
on Y
the set of all nonnegative extended real-valued continuous functions
which are finite on
Y X where
f
vew K(X,Y)
For
f
in
F(X,Y)
f e
F(X,Y)}.
we let
Af
be the set in
We set
{C(Y%Af):
X is dense in each
X we can
X.
is infinite.
K(X,Y)
Since
We denote
YAf,
by restricting the functions in
as a sublattice (and also a subalgebra) of
K(X,Y)
C(X).
to
ORDER TOPOLOGY FOR FUNCTION LATTICES
LEMMA i.
F(X,Y)
g
Given
g
K(X,Y)
in
C(Y’Ah).
is in
and
g
K(X,Y)
in
f
there is a function
in
g < f.
such that
PROOF.
that
For each function
291
-I
An
h
Bn
h
h > 0
there is a function
F(X,Y)
in
such
We consider the compact subsets
[n-l,n]
-I [0,n-2]2
h-l[n+l,
=]
1,2,...) with the understanding that B I h-I [2, ]. Using
separating functions on Y, one can construct for each n a continuous function
(n
of
Y
fn
such that
f (x)
n
h(x)
f (x)
n
and
Y
On
{g(z): z e An }
sup
the function
fOc)
sup
x
for
Thus
f
defined by
f
{in(X):
1,2,...}
n
x
Y"
in
For Y
is
_realc..ompac.t i__n Y
/
X
X
n
Moreover, f > h
x
for
X a dense subspace of
).
in
{YAf:
8X.
is realcompact in
is realcompact, since
Y
If
we will
F(X,Y)}
f
[i0].
X, we note that
compactlflcatlon of
Y,
if
This concept has been considered by Lorch in
if
f
x
F(X,Y), and g < f.
a compact Hausdorff space and
X
Stone-ech
there is a neighborhood of
Y(i.e., f(x)
and hence extends continuously to
is in
say that
An
in
is the supremum of finitely many functions
f
Y
on
x
Bn
in
is continuous, since at each point
on which
for
X
X
Where
is realcompact if and only
is realcompact in
is a quotient of
8X.
8X denotes the
Y
it follows that
X
On the other hand, the real llne
X with its discrete topology is realcompact but not realcompact in its one-point
compactification
Y:
The set
Af
is empty for each
f
in
F(X,Y)
since
X
is
not o- compact, implying
{YAf:
f e
F(X,Y)}
Y.
We note that the following proposition is also a consequence of work done in
W.A. FELDMAN AND J.F. PORTER
292
A completely regular space X is realcompact in each of its
PROPOSITION i.
compactiflcatlons if and only if it is
PROOF.
p
YX.
hx
on
X c X}
Suppose X
Y
[9,
hx(x)
such that
{hn}:=1
Let
strictly positive on X
is realcompact in
K
compact set
Y.
K in Y
image of
h
Conversely, if
co
X
is not
showing that
Lindelf, by [9] there
obtained by identifying the points of
TO will denote the order topology.
Y’X,
is a
D[X.
K.
X
Let
Since the
X is not
if
izatlon, noting that
Let
THEORE 1.
Then in
K(X,X)
To
(X)
Cco(X).
Cco
is
(in X).
We provide the following general-
C(X).
Y be a compact Hausdorff space and
K(X,Y) the order topology coincides
convergence on X
C
X is not realcompact, we conclude that
(x)
cT (x)
Is realcempact
if and only if
For a completely regular space X
X, as noted in the introduction,
o
PROOF.
Al/h,
is in
which is
will denote the topology of compact convergence and the
Cco (X) Cco (X)
Y.
p
Thus
p.
which is not contained in a zero set in
X
with realcompactlflcatlon
of
C(Y)
a non-negatlve member of
cannot be contained in a zero set in
The subscript
Since
I hn 12n,
Y.
realcompact in
subscript
we define a Urysaln function
-1
< 1. Then
0 and 0_< hx_
(,oo):
{hx
hx(p)
and zero at
be that quotient of
in X
X having a countable subcover corresponding to
’X
in
X and
is a compactlflcatlon of
for each x
1,
is an open cover of
functions
Y
Lindelf, Y
is
Arguing as in
Lindelf.
if and only if
X
X be a dense subspace
with the topology of compact
is realcompact in
Y.
Setting
z-Ct{A: fe(x,)},
we note that
K(,X,Y)
We abbreviate
K(E,Y)
K(Z,Y).
and
F(X,Y)
to
K
the relative uniform convergence structure.
and
F.
The subscript
p will denote
To complete the proof, we show that
ORDER TOPOLOGY FOP. FUNCTION LATTICES
293
{fa}
is the topology of compact convergence on Z. Let
o
a net convergent to zero in K
(As noted in the Introduction, To is the
the topology of
be
p
K
0
finest locally convex topology in which every net convergent in
There is a
g > 0
K
in
Iful
{_.ln
Clearly
Since
C
g} converges
(Z)
CO
_
n,
such that for all
< i
__.g
( >
%1.
C
to zero in
(A)
g
co
C
into
o
co
(Z)
To show that
is continuous.
co
(YAf)
C
Since
(Y)
CO
and
C
is
U
CO
(YA I)
{g
U
and
II
K:
C
To
gll
G
a support set for
Y
in
G
restriction to
assuming
U
vanishes.
does not contain
U
(a)
U
Let
F
C
[ii].
f
if for
(if
U.
be a support set for
G
< /2
then
f
To see this, consider the function
vector lattice,
0, 2g
the convexity of
is clearly in
g
is in
U;
since
U, f is in U.
F,
This follows from
To
o
F, f
in
is in
U
is a support set for
Y
F
U
We will call a compact
U
and,
K by Lemma I), the
U
then
whenever its
We note several properties of support
which will be needed.
[[fll G
[]2g[[G
f
is a fixed positive number.
Z, and
For example,
empty set is not a support set for
sets for
be a closed,
}
Z <
The following argument uses techniques found in
set
is
is solid
"denotes the supremum on
where
(Z).
(YAf) into
(g-Af) is Cco(YAf.
from
hen.ce
(As noted in the introductlon,
is continuous.
CO
o
C0(YAf)
the fact that the map from
o
is continuous.
into
To
(Z).
We remark that for
absolutely convex, solid neighborhood of zero in
C
U
let
co
C
converges in
coarser than the topology of compact convergence on Z,
the inclusion map from
C
nd hence in
{fe}
is a topological vector lattice
Thus the map from
converges.
is in
g
F.
U.
If
f
is in
F with
U.
(f
/2)V0.
Since
l[2(f-g)[[ Z
<
,
Although-F
2(f-g)
is in
is not a
U.
Thus by
294
W.A. FELDIJ AND J.F. PORTER
(b)
Let
G
F,
in
h
U
is in
G,
neighborhood of
f e F
To see this, suppose
U.
is in
To see this, let
G
g)
IIgll G
such that
fll G
U
is in
G.
G/ H.
0
If
Then
f
U
(d)
f
f
U
U, f
f
S
F
in
is a
in
and let
H’--W.
in
is in
Ko of the
C (YAf).
co
f
U.
g
0, 2[(f -g) v 0]
Now suppose
f
f^nl
is
not
{f nl}
^
bounded functions
Each
C(Y)/ F
in
Since
IIH
g)v 0
F vanish
in
U
is in
since
U
is
Y’-W
is
and
of all support sets
is a support set for
To see this, let
U.
vanish on a neighborhood
W
of
S.
Since
compact it is covered by the complements of finitely many support sets for
f
Thus
in
U
Z.
by(c).
Let
p
subset
D
e > 0.
Thus if
of
YZ.
be in
inclusion map from
C
co
into
Y.Ah such that U
g
+
f^nl e U.
U
h
in
U
is contained
F.
and vanishes on
not containing
< 6/2).
since
U
f^nl
D,
is in
is closed that
p.
Since the
there is a compact
C(YAh):[[gll D
It follows from the fact that
is a support set for
for some
is continuous
o
contains {g
C(YAh)/%F
is in
of all support sets for
h(p)
Then
(YAh)
S
For any f in F which vanishes on D,
D
U.
vanishes on the intersection of these finitely many support sets and is
We prove that the intersection
in
2g
Thus
U.
is in
The intersection
for
U
l(f
and since
is the limit in
this sequence converges to
closed, so that
G.
f e U.
is bounded there is a
f
g(x) > f(x) for x
and
Thus by the convexity of
bounded.
The function
we obtain
be support sets for
0, 2(g ^f) is in U,
U.
U.
is a support set for
U.
H
and
W of
on a neighborhood
is in
vanishes on a
The intersection of two support sets for
support set for
lg
G
h
vanishes on such a set
2(f
Since again
(c)
h
whenever
then
If for
6/2)v0 vanishes on f -I (-/2,/2), a neighborhood of
(f
g
Y.
be a compact subset of
< e) for some
g
then
is in
U.
C(Y’)
we have
f
U.
We conclude that
is in
S
Thus
is contained
295
ORDER TOPOLOGY FOR FUNCTION LATTICES
Let
some
h
enough so that the closed set
Y
h-I[N,).
h on
that
f
is in
Since
f
U.
F
_> gl
in
Y’
k
in
C(.Y-)
U_ {g
which equals
on
Y
and
f > h
on
h-l[N,).
Since
f
g
e K:
U.
is in
N
on
gl
,
Since
Igl
S
on
we conclud
S,
on
for
large
S.
is disjoint from
kvlg
is solid,
C(Y)
is in
There is a number
be
U
and
_> gl-
g
f
Letting
since
Then
h-l[N, )
is normal there is a function
and
< /2"
By Lemma I we can assume h
F.
in
llglIs
K having
be any member of
g
f
is in
Thus,
llgll S <6/2},
a neighborhood of zero in the topology of compact convergence on Z.
This completes
the proof.
3.
THE ORDER TOPOLOGY FOR A 2-UNIVERSALLY COMPLETE LATTICE
We recall that an element
weak order unit if for each
v
V{v
ne:
We will assume that
in
V
V
be the set of lattice homomorphlsms
on
V
such that
We map
V
into
topology of polntwlse convergence.
x
V.
(x)
x(v)
for all x
in
X.
V.
separate the points of
x(e)
C(X)
with the
i
by the usual Gelfand
The proof of the following proposition uses the techniques of Lemma 2 in
PROPOSITION 2.
PROOF.
Suppose
The Gelfand mapping of
@
0
for
For each lattice homomorphlsm
v
in
on
V
V
C(X)
into
V; thus,
either
x(v)
(e)
is
0
0
for all
or
(v)^n(e)
x
/#(e)
in
is in
0.
Since the lattice homomorphlsms separate
v- V{v^ne: n--
We will
henceforth
1,2,...}
identify
X, v^ne
C(X)
n.
Thus
--0.
V with its image in
as a function lattice with i (the image of
between functions in
0 for all
e).
C(X)
[3].
inJectlve.
so that
(v^ne)
X
We let
as the carrier space of
X
We will refer to
and
e
is a vector lattice having a weak order unit
that the real lattice homomorphlsms on
map
is said to be a
1,2,3,...}.
n
V
v
V
in a vector lattice
e
and refer to it
We also will not distinguish
and their extensions to functions from the
X.
X,
g.A.
296
Stone-ch
FELDMAN AND J.F. PORTER
X
compactlflcatlon
of X
to the extended real numbers.
In a function lattice V
We will have need for an additional condition.
{Vn}:. 1
we will aay that a collectlon
with 1
k
indices
2.
for each
We will say that
V
X
in the carrier space
V
of
there is a vn
O.
Vn(X)
such that
if
n, and
distinct from
x
2-dls_olnt_
is
is 2-universally
c._omplet e if each 2-dlsJolnt collection
V.
has a supremum in
For what follows we recall that a countably vnlvsally Gomlete lattice
is one in which the supremum exists for each collectlon
PROPOSITION 3.
(b)
2-universally complete.
is
lattice with
PROOF.
of
(a) The lattice C(Y)
i
with
Ifn(x)
> 0
in
C(Y),
for all
supremum of the collection
Each countably universally complete function
Y
{fn}
The proof of
in
(b)
C(Y)
y
at any point
in
Y
in a neighborhood
x
{f }
n
on
thus
N
N
is the realcompactificatlon
Given a 2-disjoint
is realcompact.
one concludes that the_pointwise supremum of
supremum of
Y
is 2-universally complete.
Y we may as well assume that
{fn}
satisfying
for any completely regular space
For (a), since the carrier space of
collection
{Vn}n= 1
there is a function
of
y.
fn
The pointwise
involves at most three functions;
{fn }
is continuous, and thus the
C(Y).
follows from the observation that a 2-disjoint
collection can be decomposed into three collections, each having a supremum by
countable universal completeness.
We note that
CR)
(R the reals) is 2-universally complete but not
countably universally complete:
vanishes off
collection
[n+ --]
{fn) n
fn
Letting
and has value
[fn[
i
be a continuous function which
at some point
fj[
Xn,
J
we obtain a
whose supremum f
0 for n
I satisfying
would clearly vanish on (-, 0) and yet f (0)
lira f(xn)
fn(Xn) i.
>-
n-
ORDER TOPOLOGY FOR FUNCTION LATTICES
X
For a function lattice V with i, we let
297
denote the quotient space of
the Stone-Cech compactification
X
of the carrier space
the equlvalance relation
if
v(p)
X
8X.
K(X, SX).
Since
Of course, if
V+
V
v(q)
8X)
V separates
is compact and (since
compact in
q
p
is contained in
for all
F(X,X),
v
V.
in
The space
Note that
Hausdorff.
V
X
is real-
is a sublattice of
K(X,)
8X then
separates
X which is induced by
K(X, SX)
C(X).
We will provide an example of a 2-unlversally complete function lattice
with
.
C(S).
which is not uniformly dense in any function space
i
such examples see
X # 8X
[7].
Furthermore, for the carrier space X
K(X, SX)
and
For other
in this example,
For this purpose we will need the following lemma.
We recall that a $-algebra is an archimedean lattice-ordered algebra over
the reals with identity
i
which is a weak order unit.
Let V be a -algebra.
LEM}I 2.
Thn every lattice homomorphism on V
is
an algebra homomorphism.
PROOF.
Let
8
homomorphism on the order ideal
element in the continuous dual
g
in
I(i)
we have
8g(f)
< 8(fg) < 8(f),
I(i)
8(f)8).
and
I(i)’
f
8 (f)8(g).
generated by i, hence an extremal
of
8 (f)
I(i)
For
in the order unit topology.
8(fg).
it follows that
Since for
0 <
8g_< 8
f > 0
in
I(i)
Thus
8(g),
so that
Th/s argument can be extended by standard means to show that
in
V.
I(i).
We next consider nonnegative elements
To facilitate computations if
the third step being valid because
8 (fg)
is a lattice
Then
l g which can be easily evaluated to be
is an algebra homomorpkism on
in
I(i)
0 < g < i, we define
for a scalar
8(fg)
V.
be a lattice homomorphism on
f
^ nl
and
8)
g* are in
0, we let
I(i).
Thus
The argument can now be repeated without the restriction
g
W.A. FELDMAN AND J.F. PORTER
298
that
g
(/).
is in
The standard extension establlshes that
V.
homomorphism on
The -algebra
EXAI@LE 1.
of all real-valued measurable
It is known (see [73) that
complete in the uniform topology.
C(S).
as a -algebra to any function space
C(S).
not isomorphic as a lattlce to any
C).
lattlce of any
f
X
Thus
C0XAh)
is in
C
topology of
co
would
8X.
To show
for some
(X)
h
[0,1]
not isomorphic
is
is not a uniformly dense sub-
X
separated the points in
C(X)
be
=
then the
by the Stone-Welerstrass
K(X,X),
F(X,X).
in
f
consider
Since
K(X,).
in
8X’Ah is
Then
Q-compact, the
By the Stone-Welerstrass theorem there
is metrlzable.
exists a sequence of functions in
on
J is
as a -algebra is closed under inversion, this would imply
But since
CX).
[0,13
It follows from Lemma 2 that
Thus
Furthermore, if
space of bounded functions in
functions on
(hence, 2-universally complete) and is
is countably unlversally complete
theorem.
is an algebra
0
f
convergent to
in
C
co
(SX).
f
Thus
is a pointwse limlt of a sequence of measurable functions, and so in
[0,i]
We remark that X here is
Just
consists of
in the discrete topology and
8X which are
those continuous extended realalued functions on
finite on a dense subset.
K(X,X)
there is a function
is a function
<_ v <_
i.
separates
f
in
V
such that
,
,
g
<_
v
in
p
Given
X
in K
1
and x
in
v
in V
x
0 < vx (p) < 1 < v (x).
x
0 < v (q) < 1 < v (y)
x
x
for all points
q
N
x
.
in some neighborhood
of
to a finite subcover
x.
{Nx}
Let vp
of
U
x
of
K
1
V
and
O < v (q) < 1 < v (y)
p
P
in
X
them
is a vector space and
such that
p
and all points
be the supremum of functions
Then
in
and satisfies
one on
since
there is a function
Clearly,
neighborhood
V which is zero on
g
f.
We begin by showing that for compact sets
PROOF.
0
For each function
V be 2-unlversally complete.
Let
LEMMA 3.
v
x
y
in some
corresponding
299
ORDER TOPOLOGY FOR FUNCTION LATTICES
for all
y
and
in
vp
inflmum of functions
Wp
in a neighborhood of
q
of p.
W
Letting
corresponding to a finite subcover
{Wp}
be the
K
of
I we
obtain
< w(y)
0 < wCq) < i <
for all
q
y
in
K2
in
uo
and some real number
A 1
F
The function
(w-l)vO]}
e,.ll
Now let
has the desired properties.
there is a function
h
F(X, BX)
in
g
K(X. BX).
be a function in
h
such that
_>
By Lemma 1
Consider the compact
g.
subsets
h-l[2n-2, 2hi
h-l[0, 2n-3] %2 h-l[2n+l,
A
Bn
and
of’
8X (n
1,2,...)
=]
with the understanding that
h-l[3,=].
B1
from the first part of this proof that there is a function
0 < vn < 2n
2n
which has value
on
An and is zero on
2-dlsjoint collection, its supremum is a function in
h
Bn
V
in
Since
with
{vn }
is a
greater than or equal to
g).
(and hence
u
Given
V
v
n
It follows
+,
V
in
[u] V
{v
we consider the ideal
V:
Iv
and we set, for each v in
%u
for some
in
}
[u],
!]
+/-f{ > 0:
< u}
It is easy to verify that the normed spaces
{([u3
v,
II
flu):
v+}
u
form an inductive system ordered by inclusion, whose locally convex inductive
limit is
VT
(see, e.g., [13,p.122]).
o
Let
PROPOSITION 4.
and having carrier space
VT
PROOF.
factors
u.
B be a 2-unlversally complete function lattice with I
X.
V
o
Then
CX,
We recall that
{u]
v, II
flu):
VT
u
{u] K,
is the locally convex inductive limit of the
eV+}, where
It follows fr L 3 that
limit of t factors
8X).
o
II
[u]v is the ideal
(X, )
flu):
u e
s
in
V
generated by
the locally convex inductive
V+}, wre
[u] K
is the
W.A. FELDMAN AND J.F. PORTER
300
K(X,X)
ideal in
V
generated by
V K (X, SX).
o
is finer than that of
o
V
To
neighborhood of zero in
For the converse, let
denotes an order inteal :In
K(X,X),
we let
{u [-u,u]V: u
42 {u [-u
uK:
V}
u
7.
i--1
V.
Denot:lng by
[-u,u] K
v--ilifi
Since
IX i I__< 1
and
in
fi
n
ui[-ui,ui ]K,
ils=n lxl-Iu
By the solidness of
U,
[-u,u]V
where
the order interval in
i
for scalars
satisfying
then
u.
ul
Ivl_<
be a solid
V} of positive
u
V},
U
of the convex hull of
V
b in the intersection with
v
{u:
Then for some collection
contains the convex hull of
U
scalars,
It is easy to verify that the topology of
u.
U.
v
The next theorem is a consequence of Proposition 4 and Theorem i.
THEOREM 2.
Let
TO
The order topology
on
is the topology of compact convergence on the
V
V.
carrier space of
4.
be a 2-universally complete function lattice with i.
V
CONVERGENCE STRUCTURES RELATED TO UNIFORM CONVERGENCE
In this section we will be using the ideas of convergence space theory (see,
e.g., [i]).
K(X,Y), where Y
We will consider convergence structures on
X
compact and Hausdorff and
is realcompact in
Y.
We let
K(X,Y)
is
denote the
.convergence space inductive limit of the system
{C (Y.Af): f e F(X,Y)}
co
together with the continuous inclusion maps.
Ko(X,Y)
in
e0
some
and
@
filter
if and only if
{ge}u
converges to
hood filter at
K .(X, SX)
such that
A e @
g
u0
g
in some
is in a factor
converges to
A
is bounded.
< w
in
Ko(X,Y) if and
factor Cco(YAf).
CI (X)
W
in a vector lattice
a
g
in
is the convergence space
A set
W
>
ge
for all
a
Wb
converges to
only if @
contains the neighbor-
We note that for
X
realcompact,
[2]
is bounded if there is an element
in
A.
g
C(Y\Af) for all e beyond
Cco(Y%Af); equlvalently, a
studied in
A filter
It is easy to verify that if
lattice then the space
{ge}
Thus a net
W6
0
w
in
is bounded if some set
is a convergence vector
containing only the bounded filters from
W6
is
301
ORDER TOPOLOGY FOR FUNCTION LATTICES
also a convergence vector lattice.
converges to a function
KG(X,Y)
it converges in
K CX,Y)
{fa}
(A net
lattice.
and an index
Kob 6X, Y)
Thus
f
is a convergence vector
in
Kcrb(X,Y)
if and only if
and is bounded- i.e., there exists a function
fel
such that
s0
<_
_>
for all
g
g
in
s0.)
We note that relative uniform convergence on a vector lattice is a convergence
vector lattice structure.
THEOREM 3.
Let V
be a function lattice rlth i.
complete, the identity map from V
PROOF.
n
1,2,...,
{v}
Thus
{vu}
Iv n
converges
u for
It follows that
Kb(X,X).
For some
>
[V]
and
<__
in
g-l[l,n]
Ire(x)]
For
x
{v}
not in
where
g
<
g-l[l,n],
By Lemma 3 there is a w
g
V
in
and
is the carrier space of V.
and hence in
Kb(X, SX).
converge to zero in
we can also assume
Thus, given n, there is an
for
g2(x)
g2(x)
V
V;
is in
g2(x)
since
1
in
(-A u)
co
u
>_ tO.
for
i
< 1
n--n
C
V
in
X
is blcontinuous.
s0.
Cco(XAg).
converges to zero in
x
CoXAu)
converges to zero in
By Lemma 3 we can assume that
for
.
n
to zero in
K(X,)
in
Kb(X,SX)
V0:
Conversely, let net
g
is 2-unlversally
onto its image in
O
converge to zero in
For some
Let net
{ve}
V
If
>
g
>_ I
n
and
{v}
such that
>
n
n g(x),
> 0
for
such that
w
>_
g2.
Thus for
u
beyond
s0
and
I w.
< nl g2 <
Iv
We conclude that
{v }
converges to zero in V
COROLLARY i.
For realcompact
C (X)
Kob(X, SX)
0
We recall that a set
if every element of
W
A
X,
CI, b(X).
in a convergence vector lattice
is the limit in
of a net in
complete if every Cauchy net (filter) converges.
If
W6
A.
W6
is dense in
The space
W
6
is complete, it
is
W
302
W.A. FELDHAN AND J.F. PORTER
follow readily
that
is
Wb
eomplete.
Let V be a 2-unlversally complete function lattice with
THEOREM 4.
Kob(X,X)
having carrier space
X.
Then
Moreover,
Vp
is complete if and only if it equals
subspace.
PROOF.
CcoOXAf),
f
Given
K,X),
in
f
f
to
a
C(X
is a sublattice of
Cco( Ag)
in
C(XAg)
is in
and separating the points of
w in V w/.tb_
as a dense
Kb(X, BX).
being an inductive limit of complete factors
is easily seen to be complete;
C(X Ag)
V
K(X,X),
The space
V
is complete and contains
i
X Ag,
thus
gb(X,)
for some
Ag)
g
F(X, X).
in
Since
containing the constant functions
there is a net
{v}
by the Stone-Weierstrass Tlore.
w Ifl {(v^w)v(-w)}
is complete.
converges to
f
V
in
converging
By Le 3 there is
Kob(,X,I"’).
in
The
last statement of the theorem is now a consequence of Theorem 3.
It follows from Theorem 4 that if a 2-unlversally complete function lattice
X
V with i separates
X
V
and if
is complete, then
V
C(X).
K(X,X) C co (X)
implying Vp
CcobCX).
is -compact and locally compact then
for some
f
F(X, SX)
in
COROLLARY 2.
[0,1J
C(X),
The space
If, moreover,
BX
since
X
Af
of all real-valued measurable functions on
with the relative uniform convergence structure is complete.
PROOF..
It was sh in Exale i that
=
K(X,).
We cite two examples to show that "relatively uniformly complete" and
"2-unlversally complete" are independent concepts.
A!PLE 2.
Let
V be the space of continuous functions
line such that the restriction of
many line segments.
that
V
V and let
is a function
of
V.
V
fy
in
G
fy] (z)
Thus,
To see
is a function lattice containing i.
G
[fyl (Y) > O,
such that
.
be a 2-disjoint collection of
K be a compact subset of
> 0
on the real
to any compact set consists of finitely
f
is 2-unlversally complete, let
functions in
X
Clearly,
f
since
For each y
in
K
is in the carrier space
y
for all
z
in some neighborhood of y.
finitely many such neighborhoods cover
K
there are finitely many functions
whose supremum
f
is positive on
K.
It follows that
there
gl
f
# 0 for
Since
at most
fyl
ORDER TOPOLOGY FOR FUNCTION LATTICES
finitely many
g
g(K)
G; i.e.,
in
303
for all but finitely many
0
G
Thus, since on any compact set the polntwise supremum of
G
finitely many functions,
which vanishes outside the interval
V;
functions in
(0,I)
G.
in
is a supremum of
However, any continuous fnctlon
is a relative uniform limit of
V is not relatively uniformly complete.
thus
LE 3.
V.
has a supremum in
g
"ultimately a polynomial"
We rlll call a function on the reals
if it is continuous and is equal to a polynomial on the complement of some
[-n,n]
interval
.
functions which are ultimately polynomials.
X of V
is
(R)
C
Where
V separates
V
D x D
f
R
x, we will prove that
since
f must be finite on
in
convergent to
p.
we would conclude that
contradiction.
{fn}n=
X.
f)
f(p).
converges to
V
On the other hand, if
{re}
Clearly
{v}
is Cauchy in
C@R).
C
co
R).
were rea]
in
chosen so that
x
and equal to
/-disjoint
e
w
V,
in
V
Thus
It follows that for all
n
{re}
_>
w
Let
fn (x)
s,8
V, there
>_ 7n (n-1,2,...
is bounded on each compact set,
converges in
7n
a
V with no supremum in
collection in
such that for
and since
R,
on the interval
is a relatively uniformly Cauchy net in
function
is bounded in
f(p)
{ru}
X
a net
is not 2-unlversally complete.
(2n-3,2n+l)
is a
If
by uniqueness of the limit of
p
{fn }
Clearly,
If so, then
RkR and {re }
be a point in
{f(ru)}
We can now show that
is a strlctly positive
in
p
whose restriction to
R.
is infinite on
be a collection of continuous functions on R
[2n-2,2n].
{ve}
f
Let
Then
is zero outside the interval
V.
R
denote the extended real-valued function on
f(x)
is
X, we can assume (examining the
.
adJ olnt maps)
Letting
We will argue that the carrier space
_C C)
V f CR) separates
and
of the set of
is the space of hounded continuous functions on
c R) C_
Since
C(R)
V be the solid hull in
and w let
C
co
(R)
to some function f
304
W.A. FELDMAN AND J.F. PORTER
is in
V,
and
{va}
converges relatlvely uniformly to
f
in
V.
V
Hence,
is
relatlvely uniformly complete.
References
1.
E. Binz, Continuous Convergence on C(X), Lecture Notes in Mathematics 469
CSprlnger-Verlag, Berlin 1979).
2.
E. Binz and W. Feldman, On a Marinescu structure on CO[),
Helv. 46 (1971), 436-450.
3.
W. A. Feldman and J. F. Porter, A vector lattice topology and function space
representation, Trans. Amer. Math. Soc. 235 978), 193-204.
4.
L. Gillman and M. Jerison, Rings of Continuous Function (D. Van Nostrand,
Comment. Math.
Princeton 1960).
5.
H. Gordon,
6.
E. Hewltt, Linear functlonals on spaces of continuous functions,
3 (50), 6-9.
7.
M. Henriksen and D. G. Johnson, On the structure of a class of archimedean
lattice-ordered algebras, Fund. Math. 50 961), 73-94.
8.
D. Hsieh,
9.
K. Kutzler,
Relative uniform convergence,
Math. Ann. 153 (1964) 418-427.
Fund. Math.
Topological properties associated rlth the A-map, Bull Aca d,
Polon. Sci. Set. Scio Math..Asronom._.__.__ Phys. 2__0 (1972), 1-6.
Eine Cha.rakterlslerung
(1975), 214-221.
yon
Lindelf-Pumen,
Arch. der Math.
2__6
i0.
E. R. Lorch, Compactification, Baire functions, and Daniel integration,
Acta Sci. Math. 24 (1963), 204-218.
ii.
L. Nachbln, Topological vector spaces of continuous functions,
40 Cl954), 471-474.
12.
AoPeressini, Ordered Topological Vector Spaces, (Harper and Row, NY 1967).
13.
H. Schaefer, Banach Lattices of Positive Operators, (Springer-Verlag, Berlin
1974).
Proc.
N.A..S: