Internat. J. Math. & Math. Sci.
VOL. ii NO.
(1988) 205-207
205
MODIFIED WHYBURN SEMIGROUPS
BETH BOREL REYNOLDS and VICTOR SCHNEIDER
University of Southwestern Louisiana
Lafayette, Louisiana 70504
(Received September 29, 1986 and in revised form November 20, 1986)
Let
ABSTRACT.
Y
X
f:
Conditions are
be a continuous semigroup homomorphism.
X u Y .is a topological semlgroup, when the
given which will ensure that the semigroup
X u Y.
modified Whyburn topology is placed on
KEYS WORDS AND PHRASES. Whyburn consruon, topological semigroup.
1980 AMS SUBJECT CLASSIFICATION CODE. 54AI0, 22A15.
I.
INTRODUCTION.
(X,m I)
Let
Y
and
as follows:
y e Y.
m
be semigroups and let
X
(X u Y,m)
Z
Let
Y
and
are Hausdorff semlgroups and that
X
Z
and
is open in
iff
Z
V n X
K
Z
and
compact, Hausdorff semlgroups, (Z,m)
is a compact map
m
in
Y
and
with
V n Y
V n Y, f
X
f
and
is
sum) topology.
X v y
Whyburn’s unified topology
X
and
Y,
is compact.
If
are open in
-I (K)
Y
is Hausdorff, and if
is a compact map, then
provided
m2(f(x),y)
x e X
if
denote this Hausdorff semigroup.
locally compact, then
f
2
and
is continuous in the disjoint union (or direct
m
respectively, and for any compact
If
Y
on
denote the disjoint union of
[I]; i.e., V
be a semigroup homomor-
X /Y
f:
may be defined on the disjoint union of
m
X, m
on
m
is
If we assume that
continuous, then
Let
(Y,m 2)
and
An associative multiplication
phism.
V
X
is
Z.
is also locally compact, so is
If
are the same.
X
and
Y
are locally
is a locally compact Hausdorff semlgroup
[2].
In this paper we consider the modified Whyburn topology which is coarser than the
disjoint union topology, but finer than the Whyburn topology and ask what conditions
will insure that
2.
will be continuous.
m
MAIN RESULTS.
Let
W
denote the disjoint union of X
topology; V is open in W iff V n X and
respectively, and
f
-I
(y)
V
f:
relative to
X
X +Y
iff
with the modified Whyburn
are open in
[3].
is closed in
y
A subset
X
A
X
is locally fiber compact.
If
Y
X
of
X
X
X
Y.
-I
A n f (x)
and
Fiber compact subsets of
V
in
is locally fiber compact iff every point in
a fiber compact closure.
Hausdorff if
A
V n Y
is compact for every
notions and facts are due to Stallings
y e Y, and
Y
and
and
Y,
The following
is fiber compact
is compact for every
has a neighborhood with
are closed in
W
W
and
is first countable, then
Z
is
and
W
B.B. REYNOLDS and V. SCHNEIDER
206
f
are the same iff
is closed.
The proof given in [2] that m is a continuous operation on Z did not use the
-i
(K) is compact for every compact K in X, but used an
assumption that m
The appropriate generalization of that condition for
equivalent condition instead.
W
is:
CONDITION I.
X
in
For every fiber compact
such that for all
x, y
X, if
g
K
THEOREM i.
Condition i, then
PROOF.
(x,y)
2)
x e X
where
W
U
V.
U
2
V
Then
y, respectively.
i
Since
K
and
x
K
y
are in
W
is closed in
Y
Since
K
Y
is regular and
Let
W.
Hence
V1 V2
maps
m
[0,I]
{(I,I- )}
(I,I)
X
[0,I]
V
Let
U
K
K
K
and
f:
X
Z
is
[0,I]
2
W
f(x,y)
by
[0,I]
{(
but
(x,y)
V.
into
Y
the
and so
open set containing
is an
K
K
V
W
is closed in
x
y.
X
If
and
with the usual
,I-)}
W
since
does not converge
W.
and hence closed in
is changed to be the usual multiplication in the
Y
is given the zero
are satisfied.
Hence
Z.
is not the same as
cally different from
satisfies
m2(f(x),y).
first factor and the zero multiplication in the second and if
W
X.
is continuous, there are Y-open
m
2
y, respectively, such that
multiplication, then the conditions of Theorem
closed map,
X
m
is regular, there is a Y-open set
Y
V
in
X
since it is a fiber compact set in
If the multiplication on
2
2.
We will show continuity
Z.
m(x,y)
w
However, the multiplication is not continuous on
and
K
is a Hausdorff semlgroup.
is fiber compact,
have the usual multiplications, then
multiplication.
{(,I)}
2
K
y
and
f-l(U i) u U i, i 1,2, are W-open neighborhoods of
f-l() V is fiber compact, Condition guarantees
V,
K in X such that if ml(x,y) are in f -i()
[0,I], Y
(0,I]
X
and
K.
and a calculation shows that
Let
Since
l(x)
of
K
g
are fiber compact for each
1,2
i
W
Y.
E
Since
y n V.
U
U
existence of a fiber compact
then
w.
containing
and
y
and
x
are the projections on
P2
is continuous and hence
such that
y
containing
neighborhoods
and
and
Pl
is locally fiber compact,
m
be an open set in
U
X
ml(x,y
K I, then
E
The argument is similar to the one given for
at a point
m2(U
X, where
in
If
X, there is a fiber compact
in
Pi (ml-l(K)),
This condition is equivalent to:
fiber compact
K
W
f
Since
is not a
is a Hausdorff semigroup topologi-
[0,I].
These examples illustrate how difficult it is to have
In
W.
continuous on
m
fact, we have:
THEOREM 2.
If
compact.
(W,m)
multiplication.
PROOF.
X
Suppose
Let
.
t,y
-I
f (t)
A,
A #
m(,y)
z.
Since
{V i}
Y
Also
f-l(y)
and let
A
m2(t,y).
z
is closed in
X
is not compact,
{yl}
in
f-l(y)
A, we can find a sequence
{x i}
Let
since
Y,
in
f
-I
A
(y)
m(A,y)
converging to
x.
is not
Y
has the zero
Xlm(x,y)
{x
z
y
z}.
Since
implies that
f-l(y)
is a limit point of
y
be a countable neighborhood basis at
contained in
y
is a first countable, Hausdorff semigroup, then
and so there is a sequence
and
is connected and for each
W.
in
Let
If we assume that no
which converges to
x
W
in
A
x
V
i
such that
is
MODIFIED WHYBURN SEMIGROUPS
# z.
m(xiY)
Hence
converges to
f
-i
(w)
0
B
{w} u
neighborhood of
A
m2(Y,y)
that
f-l(z)
{ml(xi,Yi)}
is not in
B
B
is finite because otherwise
compact set
Thus
ml(Xi,Yi)
Thus the set
z.
z
f-l(w).
z.
m2(t’,Y)
Let
for all
B
X
z
z’
X
since
and let
z’
and
X.
{ml(xi,Yi)}
w e Y,
For any
is fiber compact and
which contradicts the fact that
t’,y’ e Y
z’. Hence
i, but
is closed in
will have a convergent subsequence in the
This means that
is open and must equal
207
{ml(xi,Yi)}
is connected.
m2(t’,y’).
Y
W
B
is a
converges to
z.
All of this yields
The argument above will give
has the zero multiplication.
REFERENCES
I.
2.
3.
WHYBURN, G.T. "A Unified Space for Mappings", Trans. Amer. Math. Soc. 74(1953),
344-350.
SCHNEIDER, V.P. "Compactifications for Some Semigroups Using the Whyburn
Construction", Semigroup Forum 13(1976/77), 135-142.
STALLINGS, Y.O. "A Characterization of Close Maps Using the Whyburn Construc=ion", Internat. J. Math. and Math. Scl. 8 (1985), 201-203.