Internat. J. Math. & Math. Sci.
VOL. 17 NO. 2 (1994) 357-360
357
THE MEET OF LOCALLY CONNECTED GROUP TOPOLOGIES
DORENE J. FOX
Louisiana State University at Alexandria
8100 Hwy 71 South, Alexandria LA 71302
ABSTRACT.
(Received November 6, 1992 and in revised form March 23, 1993)
The collection of all group topologies on a given group is a complete
The purpose of this paper is to develop a
lattice partially ordered by inclusion.
convenient
method
of
defining
meet
the
of
an
arbitrary
collection
of
group
topologies and demonstrate that the meet of a collection of locally connected group
topologies is also locally connected.
KEY WORDS AND PHRASES.
Topological
groups,
quotient
spaces,
locally connected,
translatable and semicontinuous topologies.
1992 AMS SUBJECT CLASSIFICATION CODES.
I.
22A05, 54B15.
INTRODUCTION.
There has been considerable interest, by Graev [I], Comfort [2]
and others, in
finding group topologies on a given group. An approach is to start
on a given group and derive
an associated group
topology.
with a topology
In [3] Roelcke and
Dierolf developed a method for finding a neighborhood basis for
the finest group
topology contained in the intersection of an arbitrary collection of
topologies on a
group. In general, however, this basis does not consist of open sets.
In [4] Clark
and Schneider studied the meet in the lattice of group
topologies and developed a
convenient method for finding a basis of open sets for the meet
of a countable
collection of group topologies on an abelian group.
In
this
paper
we
will develop a convenient method for finding an open
the finest group topology contained in any topology on a
group. This method will enable us to find a neighborhood basis for
the meet of an
arbitrary collection of group topologies and demonstrate that the
meet of an
arbitrary collection of locally connected group topologies is
also
neighborhood basis for
locally
connected.
2. PRELIMINARY RESULTS.
Let G be a group.
Then it is well known that
is a fundamental system for G
is a collection of subsets of G each containing the
identity and satisfying
iff
the following properties.
I)
If U,V
2)
If U
3)
If U
and x
4)
If U
then there exists V e
such that V -I c U.
5)
If U
then there exists V
such that VV c U.
then there exists W
and a
such that W c U
U then there exists V
G then there exists V
V.
such that aVc U.
such that xVx -I c U.
If
is a fundamental system then there exists a unique
topology t on G such
that t is a group topology having
as a basis at the identity.
DEFINITION I.
Let G be a group and
containing the identity of G.
properties I,
Then
2 and 3 above and by [5]
satisfies properties
be a collection of subsets of G each
is a translatable system for G iff
satisfies
is a semifundamental system for G iff
through 4 above.
In [5] Clay defined a semicontinuous topology and showed that a basis
of open
D.J. FOX
358
neighborhoods at the identity is a semifundamental system.
F
She also showed that if
is a semifundamental system on a group G then there is a unique topology t on G
as a basis at the identity.
such that t is a semicontinuous topology having
DEFINITION 2.
A translatable group is a group G endowed with a topology t such
Ua and aU are elements of
that
for
U
all
t and a e G.
translatable group then we say that t is a translatable
(G,t)
If
is
a
topology on G.
It is easy to see that if t is a translatable topology then a basis of open
neighborhoods at the identity is a translatable system and if
is a translatable
system for a group G then there is a unique topology t on G such that t is a
as a basis at the identity.
translatable topology having
Let G be a group and t any topology on G.
DEFINITION 3.
The Graev operator
assigns to the pair (G,t) the pair (G,g(t)) where g(t) is the finest group topology
on G contained in t.
The collection of all group topologies on a given group is a complete lattice
partially ordered by inclusion.
topologies {t
At
[6] that
g(OAtm)ct m
define
for all A and t is a group topology}.
g(oAt)e
--
G for all
(R)rag
m. i=1
a translatable topology on
(R)G
topology inherited by
mm"
THEOREM 1.
--
m:(llG, T)
and b
Let V
Let
exists n
[[G
when
I=1
denote
HG
I=IHGI
and
Let T be
and S be the
is endowed with the box
G
1=1
then the quotient map
<e h
@G
i=1
<x>
h
and h
h
-I
4
and <a >
uVw.
such that
b b
h
’"
-I
m-1’
u<b
h
-I
>w
1+1
COROLLARY 1.
b
-1
1+2
...b a a
a
m-1 m-2
...a
,e,e,e,...> and w
T
uVw
e for all
Let n
there exists m such that a
n
since
<y>
23
(rinG,T)
---->
1=1
Let T be a
G is open.
for
all
i=3 4 5 ...m.
<hl,h3,h4,’.’,h
,e,e,e,...>
T is translatable.
V such that
Let k
max {m,n
<xl >
u<y>w
and there
then:
mooCV) m
and define m.
C2.1)
HnG
G by m (a
i=1
a a a ...a
e
123
Then there exists
Yl
G
i=1
V such that
a a a ...a
123
-I
then there exists <b >
<a > and <b > are in
m and b b b .--b
b-1 a
-I
ml(moo(V))
YlY2Y3"’’Yk
m
I=IHG
G and gm[t] shall denote the quotient topology q(S).
Since
e for all n
then u,w
1__eGl,
A
qm(T) shall denote the quotient topology on G
Then
T and <a >
m0(<a >).
u
g(oAt)=
e for all m > n.
where a
If T is a translatable topology on
Let h
Let
denote
(in general) but
----> (G,q(T)) is open.
PROOF:
mo(<b >)
A’tm =AtQ
t be a translatable topology on G
i=1
as a subspace of
I=I
(ITG,T)
I=lOG
23
(R)G
Samuel pointed out in
is a group topology. Therefore
a a a ...a
topology with t on each factor.
induced by
Therefore
then we let
G by mm(<a >)
collection of group
A} as a subbasis and the meet is the
for some
may not be a group topology.
for all A and
GI
If
t
has {V "V
join of {t "t c t
The join of an arbitrary
translatable
topology on
HnG
i=l
then
a
1’ 2’
the
a
3
.a
quotient
map
MEET OF LOCALLY CONNECTED GROUP TOPOLOGIES
359
THE GRAEV OPERATOR.
3.
Let n {0,1,2
}, P
denote -1(i)
and
all
for
P
and
i
A
(A
u
OPn
A
2
A
A0
Then {P(<A > n)" A
(R)mG.
x A
and n {0
n+2
is known as the cross topology due to
the "shape" of the basic neighborhoods of the identity element.)
C(<A >)
u
0"
,
_,(A
n
--
0
A A0 --’A0 )] t’hen (s)
G
C
qm(C ).
One can easily see that Cs) satisfies properties I,
PROOF:
c
<A >
Then V
2
Therefore
<U
Then there exists V
V
g(s) such that e
Po
Let
Then
=
V V V V
3
4
V and
and
k
l,n
=
f such that
U n U
> and V
<U U U
2’ 4 6
then:
V
I+I
Let:
(3.2)
C(<V >).
c V
1-1
)2
=
(V
1,2
=
q(C
s.
)2
3
c (V
k
3,4
3
O’n
C
We can also find
U.
2,3,4
for all
3
k
=
and (V)
min{m,O’m+l,O’m+2,
km,
)2(Vk
1,2
(V)3
= (Vk
V V
(V
such that V
g(s) such that e
2
k
I=1
U.
U,
(I
V V V
2
1,2
denotes
C
3
-I (i)
3
(V
=U
for
all
and
1,3
c U.
1,4
Suppose then that VO" V0 VO" -.-VO"
Let
WI
nnU ’.
=
U and suppose F is the set of all open neighborhoods of
2
2
,
all
then there
n and W
is a group topology and since s is translatable
q(C
g(s) such that e
).
for
find,
<B >
>
1’ U3,U s,
e.
i
ala1/la1/2’’.an
for all i=1,2,3
1,3,5
f for
[C(<AI>)][C(<Bt>)] =
Let U
can
we
and
bl
Vl
Therefore a[C(<W >)1 c U.
There exists V
{5)
Let
and we can find
3
n W
Ui
aW
a
...an
a123a
,n.
1,2,3,
Therefore we
3
2
such that b U’ c V b
U U U --.U
UB UB UB ’’’UB
such that a
n such that a
for all
and W
2
W
UB
Po
,2,3,.
for i=
c
Vial
U’
exists
Now
U
C(<U >)
U then:
and
There exists n
al
3 and 4 of a
Let U
Therefore let us consider properties 2 and 5.
be an element of (s) and a
such that
This follows from the fact that the quotient map
g(s).
q(C
fundamental system.
can find
Now if we let
f} is a neighborhood
G is open.
THEOREM 2.
(2)
{C(<A >)-A
3
2
basis of the identity for
m
(3.1)
e ...)
e
n+]
}} is a translatable system for a topology
2
(Introduced by Bradd Clark, C
1=1
A0
x
(r
n+l
]E[N
on
Let <A > be a sequence of
and:
P(<A >,n)
C
Suppose s is a semicontinuous
is a semifundamental system for s.
topology on G and
elements of
be the set of all permutations on {n+1, n+2, n+3
=
(V
3
k
1,1
=
U for all
s n
and
Po"
D.J. FOX
360
V
V
V
’’’V
V
V
V
...V
V
V
...V
j-1
)3
c (V
V
Therefore C(<V >) c U and g(s) c qm(C
)3
(V
1,j-1
k
1,n
c s.
(3.3)
J+l
3
c (V
k
j+l,n
c U.
(3.4)
1,n
Therefore g(s)
q(C
since
q(C
is a group topology
If G is abelian then (s)
{nN(V V2 V ...Vn)
V
e Y
and by a proof similar
tO the one above we can prove the following corollary.
COROLLARY 2
If G is abelian then g(s)
q[s].
Suppose then that we have an arbitrary collection of group topologies {t
on
a
given
group.
Since
In
[7]
and
Clark
e&te
is
semicontinuous
we
have
that
(e&te)
is
a
A t
fundamental system for
Schneider
defined
S(t)
to be
the
finest
semicontinuous
topology on G contained in t and gave a concrete description of S(t).
is semicontinuous g(S(t))
g(t).
Since g(t)
Therefore we can also use Theorem 2 to prove the
following corollary.
If t
COROLLARY 3.
is any topology on a group G and C
topology with respect to S(t) then qm(C
s(t)
4.
s(t)
is Clark’s Cross
g(t).
LOCAL CONNECTEDNESS.
Suppose A and B are connected subsets of a translatable group G then aB is
connected for all a e A.
Let x
B then Axc AB and Ax is connected. Therefore AB
Ax u AB
Ax u [aA(aB)] is connected since ax
A. Therefore
Ax naB for all a
we have the following theorem.
THEOREM
Suppose t is a locally connected and translatable topology on G
3.
then qm[t] is also locally connected.
In a similar fashion we can prove the following corollary.
If t is a locally connected and translatable topology on G and C
COROLLARY 4.
is Clark’s Cross topology with respect to t then
Suppose then that {t
on G then
t
qm(C
is also locally connected.
is a collection of locally connected group topologies
is also locally connected and therefore
At
g(At)
is also
locally connected.
ACKNOWLEDGEMENTS.
Dissertation,
titled
Many
of
these
"Translatable
results
Groups
were
and
the
developed
Graev
in
the
Operator"
author’s
under
the
direction of Bradd Clark, University of Southwestern Louisiana, 1991.
REFERENCES
I.
GRAEV, M. Free Topological Groups, Izv. Akad Nauk SSSR Ser. Mat. 12 (1948),
2.
COMFORT, W.W. and ROSS, Kenneth A. TopoloKies Induced bv Groups of Characters,
291.
Fund. Math., Vol. 55 (1964), 283
3.
ROELCKE, W. and DIEROLF, S. Uniform Structures o__B Topological Groups and Their
Quotients, McGraw Hill, 1981.
4.
CLARK, B. and SCHNEIDER, V. The Meet Operator in the Lattice of Group
481.
Topologies, Canad. Math. Bull. Vol. 29 (1986), 478
5.
CLAY, E. Semicontinuous Groups, Dissertation, University of Southwestern
6.
SAMUEL, P. Ultrafilters and Compactifications of Uniform Spaces, Trans. AIMS 64
132.
(1948), 110
7.
CLARK, B. and SCHNEIDER, V. A Comparison of Topologies on Free Groups, To
Appear.
279
324.
Louisiana, 1990.