I nternat. J. Math. Math. Si.
(1981)55-65
Vol. 4 No.
55
A COMPLETION FUNCTOR
FOR CAUCHY GROUPS
R. FRIC
D.C. KENT
Department of Pure and Applied Mathematics
Washington State University
Pullman, Washington 99164
U.S.A.
(Received March ii, 1980)
ABSTRACT.
A completion functor is constructed on the category of completely
normal Cauchy groups and Cauchy-continuous homomorphlsms.
A competlon functor
is also obtained for a corresponding category of convergence groups.
KEY WORDS AND PHRASES. Completion functor, Cauchy group, pre-Cauchy group,
completely normal Cauchy group, convergence group
1980 MATHEMATICS SUBJECT CLASSIFICATION CODES:
i.
54A20, 54E15, 18A5
INTRODUCTION.
In [ 2]
pletions.
we developed a general functorial approach to Cauchy space com-
In this paper, we consider completions of groups equipped with com-
patible Cauchy structures.
Such spaces have been studied by R.N. Ball [ I], who
used them to obtain certain completions of lattice ordered groups.
We shall use the definitions, notation, and terminology of [ 2].
However
the modification functor approach to constructing completion functors used in
R. FRIC AND D.C. KENT
56
[2]
is not employed here.
Although at least some of our results could be
obtained via that method, it is found to be less convenient due, in part, to the
fact that
Wyler’s completion does not preserve the compatibility between group
and Cauchy structures.
Let
PCHG
be the category whose objects are groups equipped with Cauchy
strudtures relative to which the group operations are Cauchy continuous (such
groups are said to be pre-Cauchy), and whose morphisms are Cauchy-continuous
homomorphisms.
of
PCHG.
All Cauchy space categories considered here will be subcategories
For convenience, we shall use the term "subcategory" to mean "full
subcategory", so that subcategories are completely determined by their objects.
In applying the definitions of such terms in
[2]
as "map", "completion", etc. to
our present setting it is, of course, necessary to replace "Cauchy-continuous
function" by "Cauchy-continuous homomorphism", etc.
X will represent a group with multiplicative
Throughout this paper,
operation and identity element
and x
e
F(X)
will denote the set of all filters on X,
x E X.
will denote the fixed ultrafilter determined by
are applied to subsets and filters in the obvious way.
,
E F(X),
( FG :F (3
-I
{F-I:
F
} and
Group operations
In particular, for
is the filter generated by
G (.}.
A completion functor is constructed on the category
Section 3, we consider the subcategory
of
PCHG
PCHG
in Section 2.
In
consisting of Cauchy
groups, and show that the completely normal Cauchy groups form a completion subcategory of
CHG.
Whether a completion functor exists on all of
unsolved problem at this time.
CHG
remains an
Section 4 examines the relationships between
Cauchy groups and convergence groups.
The results of Section 3 are used to
construct a completion functor on the category of completely normal convergence
groups and continuous homomorphlsms.
COMPLETION FUNCTOR FOR CAUCHY GROUPS
57
PRE-CAUCHY GROUPS
2.
(X,C), where X is a group and C a
A pre-Cauchy group is a pair
set of
X satisfying the following conditions:
filters on
(CG I)
C
(CG2)
If
(CG3)
If
(CG4)
If
{
,
C and
_<
6C
then
N
Note that if
,
X.
for all x
C
then
-I
C.
C, then x =y.
(X, C)
is a pre-Cauchy group, then
formation of inverses and products of filters.
C
is closed under the
It is easy to see that if
X
is
a separated (Hausdorff) topological group and C is the set of all Cauchy filters,
(X, C) is a pre-Cauchy group.
then
PROPOSITION 2.1
is a pre-Cauchy group, then
C
is a Cauchy
X.
structure on
and
(X, C)
If
PROOF.
The first two Cauchy space axioms (see
(CG2).
If
and
N
and therefore
[2])
are nondlsjoint filters in C, then
{
C
(CG I)
-I
are covered by
N
>_
this establishes the third Cauchy space axiom.
The Cauchy structure associated with a pre-Cauchy group is called a
"group Cauchy structure"
in
[I].
This should not be confused with our term
"Cauchy group structure", by which we shall mean the Cauchy structure associated
With a Cauchy group (see Section 3).
Starting with a pre-Cauchy group
equivalence classes and
define
[][]
As in
on
[2]
[]
j :X
and
/
X*
(X, C ), let
the canonical map.
/ x
be the set of Cauchy
Given
[5], ]
{
X*
[]-I [-I].
we associate with a Cauchy space
X defined by
X*
if
N
(X, C
the convergence structure
C (or, equivalently, if
[]).
For
pre-Cauchy groups, equivalence of Cauchy filters is characterized in terms of
convergence in the next proposition.
58
R. FRIC AND D.C. KENT
PROPOSITION 2.2
N
_
C
iff
PROOF
Let
-I
n
(-I n
Then
C
e.
/
,
N
-i
Thus
e
/
-I
,w
(X, C) be a pre-Cauchy group and
Let
-I
>_
-i
and $ >_
C
-i
Conversely,
e
/
-I
Then
$
implies
C implies
m c.
&)
PROPOSITION 2.3
(X,C)
If
X*,
is a pre-Cauchy group, then
the operations defined above, is a group, and
X
j
X*
/
equipped with
is an isomorphic
embedding.
We shall verify only that the multipllcative operation on
PROOF
I2
-i
-
K
and
e
I
fl
(X, C
the filter
for
1
[ 2 ] and [w I]
+
@
(,2,’ ,2,, -1
1
K E C
By Proposition 22,
j ()
n [].
X*
X*
Any filter on
F(X*)
C
{ E F(X*)
LEMMA 2.4
for some basic filter
^
A basic filter on
filter of the form
is of the form
^i
E
[
i ]"
X*
Let
If
e
@i
[
^}
is either a fixed ultrafilter, or else a
is a free filter and
where
A basic filter A on
PROOF.
i i i
will
be the set of filters defined by:
*=
where each
X*
obtained by taking a
finite product of subbasic filters will be called a basic filter.
C c
define
Filters of the form
and also all fixed ultrafilters on
C
be called subbasic filters on
E C
For each free filter
@
as follows:
a free filter in
/
e.
/
be a pre-Cauchy group.
X*
is
I i 2 2 )-I e.
-i
-1
2 E C If
I (I 2 -11.] ) 2
and (i
i )( 2"2 )-In & >-
e)
2
$, then
on
[2 ]"
e, and it suffices to show
II )( 22 )-I
which implies
Let
>-
I2
Note that
and
[i ]
-I
i "2
Let
well-defined
X*
X*
e (
X*.
can be written as a product
[i ]
I 2"’"
and
n
]
[i ]
@i
then
^
j
^I^2...An
i [i ]’
where
where
the intersection of translations of free filters, and hence a free filter.
is
COMPLETION FUNCTOR FOR CAUCHY GROUPS
THEOREM 2.5.
(X, C
If
C*.
(X*, C*)
(CG4)
Axiom
(X, C).
follows easily from Lemma 2.4, and so
C*
is finer than a basic filter of the form
We know already that
e.
and hence convergent to
(X*, C*)
It also follows from Lemma 2.4 that
is a pre-Cauchy group.
is complete, since each member of
0
(X, C
j
/
(X*, C*)
Cauchy-
is an isomorphic embedding, and this map is clearly Cauchy-continuous.
continuity of
whenever
J
-I
follows because
j
-I
is an isomorphism and
(X,C).
a pre-Cauchy group completion of
To show that this completion is strict, let
>_ ^.
such that
^
C
^i
[i ] j (i) [ Ni ]"
>_ cl
cl
in
^ >_
el j
(i
j
-I
(^)
C
((X*, C*),J)
Therefore
X*.
is
Then there is
where each
^ ^1^2 ..An,
Assume that
Then
-
J(X).
is a subbasic filter containing
^
is a
(CG I), (CG2), and (CG 3) follow immediately from the
The axioms
construction of
((X*, ) ,j)
is a pre-Cauchy group, then
strict pre-Cauchy group completion of
PROOF.
59
I NI’’" n n Nn )’
and the
conclusion follows.
THEOREM 2.6.
and
If
f :(X,C
(X I, C I)
/
is a complete pre-Cauchy group, then there is a unique Cauchy-
(XI, CI)
continuous homomorphism
which makes the following diagram commute:
f
J
(x*, c *)
---->
(x, c
cI
(x I
is well-defined.
clear.ly
f()
/
z.
Then
f() f()
which implies that
$
/
C
For each
PROOF.
in X
is a Cauchy-continuous homomorphism,
([])
define
Let
([])(])
f()
/
If
f()
/
f()
yz, where
([])
yz, and so
is a homomorphlsm.
then it is clear that
if f
y
+
y
in
XI;
f()/ y,
([][])
yz
is a subbasic filter and
f(u) in
XI;
the fact that
f
is
R. FRIC AND D.C. KENT
60
Cauchy-continuous follows from the fact that
follows because all spaces involved are Hausdorff.
ness of
PCHG
Let
be the category of pre-Cauchy groups and Cauchy-continuous
PG*
homomorphisms, and let
For objects
f
(X I,
The unique-
is a homomorphlsm.
f
eI
(X,C)
(
(X2,
e2
+
PC,...
be the subcategory of complete objects in
PCHG
(X*, e)-
K(X,e)
define
PC.HG,
is a morphism in
If
(X 1 * ,e
define Kf
;) (X; ,e;)
be the unique extension :rphism whose existence is guaranteed by Theorem 2.6.
Then
K
identify
PCHG +
i
k
PCHG*
If, in the notation of
is a covariant functor.
with the embedding
X
j
/
X
[2],
we
then the following corollary is an
immediate consequence of Theorem 2.6.
COROLLARY 2.7.
3.
K
PCHG.
is a strict completion funcor on
CAUCHY GROUPS.
A pre-Cauchy group
(X, C
condition is satisfied:
(CG 5)
Cauchy group if the following
-I
( C
C
is said to be a
-I
It is natural to ask whether
C and
(X*, C*)
group whenever
is a Cauchy
(X,C)
The answer is not known in general, but the answer is yes for
is a Cauchy group.
a certain class of Cauchy groups which we shall now discuss.
LEMMA 3.1.
(X,C) are
The following conditions on a Cauchy group
equivalent.
If
(CN 2)
If
6
(CN 3)
j(X)
is a normal subgroup of
PROOF.
and
(CN I)
C and
(CN2).
X, then
x
If
6
-I
--1
C and
(CN 2)
(CN I) that
(CN3).
.,
is non-convergent, then
x
e
which implies that
-I
is non-convergent.
is convergent.
,
X
are both non-convergent members of
the assertion of
-I
6 C are non-convergent, then
(CN I)
where
S S -i
,--
--1
But
x
This contradicts
+ x.
must be non-convergent.
We shall show that
e-I j (X) e
J(X)
for all
(X*
j(X).
COMPLETION FUNCTOR FOR CAUCHY GROUPS
,
If
X*,
and
X: thus
y
(CN 3)
[5]
x X, 8
.-I (8), and j(x)
]
e-I
8
X, and
x
j (X)
then
e
j(y) e
-1
-I
x
j (X)
.
-I
j(X) by
j(x)e
Se-I
is convergent, then
CN
/
for some
Y
and let
j(x)
for some
This implies that
3.
(CN2).
by
be non-convergent filters in C
Let
If
j (x)
(
If
-I
e
=
(CNI).
8
-i x]" -I
j(x)-l.
Conversely, let
y
[
X, then
x
61
is convergent,
and this contradiction completes the proof.
(X, C
A Cauchy group
which satisfies any of the three equivalent
conditions of Lemma 3.1 is said to be
Note that all complete
completely normal.
Cauchy groups and all Abelian Cauchy groups are completely normal.
THEOREM 3.2.
(X, C
If
(X*, C*)
is completely normal, then
is a Cauchy
group.
(X*, C*)
We need to show that
Proof.
-I
^I ,.. "’An
then there are subbaslc filters
-I >_
(CG5).
satisfies
Assume that
such that
Recalling the form of the subbasic filters, we can argue that
^i^2...^n
m
any finite product of subbaslc filters is also expressible in the form
where each
ij
6
C
If
(X,C)
C.
and
i, 1 _< i _< m, choose
m
AA-I
c_ U 8 i j (Gi) Yi"
i=l
and
A’
i
6
,
If
then for each
such that
A’6 -I
(
-I
(not necessarily in
e
G
ik i)
ij X*
where
m
>_
i=l
Gi’
c__ U 8 i
,
G
i
such that
i j( i i’
Then there is
G
i
(but
G
i’
can
#i
-and
8 i,
A E
such that
c__ U 8 i J(G i) Yi"
i=l
_c
and
where
If
i’
A’ c_ A
is not necessarily
From these observations
i
I _<
X*
Yi
m
A, then A -I
m
8i
where
i
..
i
there is
i
i=l
that there is at least one index
B
j( i
are as previously specified.
For each
in
i
i
Thus, we shall assume that
i
]
is completely normal, then the expression for
be reduced to one of the form
i
ikl
i2 j( i2
i =il j il )
i’
it
follow
_< m, for which there exists a filter
-I
and
j
()
are filters
R. FRIC AND D.C. KENT
62
which do not contain disjoint sets.
_
I
Let
(")
j
-i
(CGs)
I-I
(
implies that
AIA2 ...An
CLG
Let
CNCHG.
CNCHG*.
^IA2 ..An 1
>_
E
C
I
-I
>
1
which, by
*"
(
Thus
since each of the filters in the
and the proof is complete.
C
PCHG
be the subcategory of
CNCttG*
noel Cauchy groups, and let
in
which implies
fl -II-I
From this, it follows that
C
implies
latter product is in
*
(
j(
This implies that
j(
Then
determined by the completely
denote the subcategory of complete objects
It follows from Theorem 3.2 that the functor
l
maps
CNCHG
into
K
Combining this fact with Corollary 2.7, and using the se symbol
for the restricted functor, we obtain
CNCHG of completely
is a completion functor on the category
K
IEORI 3.3.
normal Cauchy groups.
In other words, the completely normal Cauchy groups form a completion subcategory of the pre-Cauchy groups.
The completion by
K of an Abelian Cauchy
group is clearly Abelian; thus we have also established
COROLLARY 3.4.
is a completion functor on the category of Abelian
K
Cauchy groups.
4.
CONVERGENCE GROUPS.
that is,
(X,q)
C
q
/
"group
X
and a con-
relative to which the group operations are continuous;
q
q
-I
-I /
With each convergence group
xy
and, / y implies
vergence structure
q
is a pair consisting of
(X,q)
A convergence group
x
q
there is associated a natural Cauchy structure
{
-I
E F(X)
Cauchy group
as follows:
(X, C
/
x
/q
e
and
-I
q_; e}.
On the other hand, every pre-
gives rise to a convergence structure
on
X
defined
(
iff
A convergence group
q
(X,q)
is said to be
st.r.0ngly normal if
-I
q
/
e,
COMPLETION FUNCTOR FOR CAUCHY GROUPS
, q+
q
-I
e, and
/
3
imp lies
e
63
q
3-I
e.
/
The definition of strong
[i].
normality and statement (b) of the next proposition are due to R.N. Ball,
PROPOSITION 4.1.
(a)
(X, C
If
(X, q
is a pre-Cauchy group, then
is
c
a convergence group.
(b)
(X,q)
If
group iff (X,q)
(c)
(X,
then
qc
(a)
Let
Obvious.
/
"y-i
N
(c)
(X,c q)
is a convergence group and
is a pre-Cauchy group,
is a Cauchy group.
Cq
PROOF.
is a pre-Cauchy
is strongly normal.
(X,q)
If
(X,c q
is a convergence group, then
qc
x
/
C
y
n
and
qcx y-i
-I
Then
/
>_ (n x)(m
(xy
I
PROPOSITION 4.2.
For any group
--
because
X, the relation C
-+
qC
is a one-to-one
correspondence between Cauchy group structures and strongly normal convergence
group structures on
PROOF.
If
X.
(X, C
is a Cauchy group, then
convergence group by Proposition 4.1.
If
(X,q
C’ =C
is a strongly normal
C
then
qc
C’
is also a
Cauchy group structure by Proposition 4.1, and it remains to show that C
-1
If
E C and since
is finer than
C then 3
C and
C’.
-I
-I
and
6
both of these product filters q-converge to e’
Conversely,
C’..
C
-I
/
by
-i
e.
Thus
if
-i
E
and
C’
-I
then
-I
qc+ e
are both in
Therefore,
and
C
and
C
follows
(CG5). I
It is obvious that Abellan convergence groups are strongly normal.
Thus
we ob taln
COROLLARY 4.3.
If
X
is an Abelian group, then
Cq
+-+ q
is a one-to-one
correspondence between Cauchy group structures and convergence group structures
on
X.
R. FRIC AND D.C. KENT
64
(a)
PROPOSITION 4.4.
:(X, C)
If
(X I,
/
C1
(X,q
homomorphlsm between pre-Cauchy groups, then
-
is a Cauchy-continuous
C
con tlnuous.
(b)
(X,q)
If
/
(Xl,ql)
If (X, C)
(c)
is
1
is a continuous homomorphism between strongly
(X, C q)
normal convergence groups, then
(Xl,qC
/
(X I,
Cql)
is a complete Cauchy group, then
is Cauchy-contlnuous.
(X,q
is a complete
C
convergence group.
(d)
(X,q)
If
is a complete convergence group, then
(X,C q) isa
complete Cauchy group.
Proofs are given only for (b) and (c), the other parts being similar.
q
q
-1 /
e
e implies
Let
Then
() (3) I
3
PROOF.
()-I
Similarly
(c)
Let
qc e.
I
(3-1)
Cq
(b)
+
and so
q+l
)
e, and so
()
"
C
ql
-I qc
+ e and
(X, C) be a complete pre-Cauchy group, and let
{
en
Cq C as was sho in the proof of Proposition 4.2,
C
is q
C-
convergent because (X, C
We define a convergence group
(X, C q
strongly nol d
(X,q)
is complete.
to be
qompletely nodal if
is a completely nodal Cauchy
all Abeli convergence groups
group.
it is
Note that
d complete nvergence groups are completely
normal.
Let
Cdenote
the category whose objects are completely normal con-
vergence groups and whose morphisms are continuous homomorphisms:
denote the subcategory detened by the complete objects in
K
CNCG
/
classes, and
CNCG* by
q*
Proposition 4.4.
K(X,q)
(X*,q *)
X*
(X*,q*)
where
,
CNCG.
is the set of
let
CNCG*
Define
C q-equivalence
CNCG*
is a consequence of
the fact that
q
Using Proposition 4.4 along with the results of Section 3, we
qC
obtain the following.
TKEOREM 4.5.
K
is a completion functor on the category
normal convergence groups.
In particular,
K
CNCG of completely
is a completion functor on the
65
COMPLETION FUNCTOR FOR CAUCHY GROUPS
on the subcategory of
CNCG
consisting of all Abelian convergence groups.
The existence of a completion functor on the Abelian convergence group
category was established in
[3]
using methods and terminology based on
[4].
REFERENCES
[I]
R.N. Ball, "Convergence and Cauchy Structures on Lattice Ordered Groups",
Trans. Amer. Math. Soc. (to appear).
[2]
R. Fri and D.C. Kent, "Completion Functors for Cauchy Spaces",
International J. Math. & Math Sci. 2(1979) 589-604.
[3]
R. Fric and D.C. Kent, "Completion of Pseudo-Topological Groups",
Math. Nachr. (to appear).
[4]
S.
Ghler,
and G. Kneis, "Completion of Pseudo-Topological Vector
Math. Nachr. 7_1976) 185-206.
Spaces",