Claim
:
Q
=
{
e
}
. Assume the contrary. Then we can fix
x
2
Q
such
that
x
6
=
e
. There exist open sets
U
and
W
such that
x
2
U
,
e
2
W
,
and
U
∩
W
=
?
. Since
xe
=
x
2
U
, and the multiplication on the left is
continuous at
e
, we can also assume that
xW U
.
Since
γ
is a
πτ
-network at
e
, there exists
V
2
γ
such that
V W
.
Then for the point
y
V
we have:
y
V
2
W
,
xy
V
2
P
V
y
V
V W
, and
xy
V
2
xV xW U
. Hence,
xy
V
2
W
∩
U
and
W
∩
U
6
=
?
, a
contradiction. It follows that
Q
=
{
e
}
. Thus,
ψ
(
e, X
)
≤
τ
.
To prove Theorem 3.2, we need one more technical result:
Propositional 3.9.
Every rotoid
X
is twistable at some point. Every
strong rotoid is twistable at every point.
Proof.
Since
X
is a rotoid, we can fix
e
2
X
and a homeomorphism
h
of
X
×
X
onto itself such that
h
((
x, e
)) = (
x, x
)
and
h
((
e, x
)) = (
e, x
)
,
for every
x
2
X
. Now we define a binary operation on
X
by the rule: for
x, y
2
X
, put
xy
=
z
, where
z
is the second coordinate of
h
((
x, y
))
. Since
h
is continuous, this binary operation is continuous. Obviously,
ex
=
x
=
xe
,
for any
x
2
X
. Hence, the binary operation so defined is a twister. The
second statement is proved similarly.
The proof of Proposition 5.1 from [6] given there proves a somewhat
stronger statement. This argument concerns a space
X
with a
G
δ
-point
a
and a homeomorphism
g
of
X
×
X
onto itself such that
g
(Δ
X
) =
X
×{
e
}
,
for some
e
2
X
. It is not assumed that
e
coincides with
a
. The argument
in [6] proceeds as follows:
“. . .
g
(
a, a
) = (
c, e
)
, for some
c
2
X
, and
(
a, a
)
is a
G
δ
-point in
X
×
X
.
It follows that
(
c, e
)
is also a
G
δ
-point in
X
×
X
. Therefore,
e
is a
G
δ
-point
in
X
which implies that the set
X
e
=
X
× {
e
}
is a
G
δ
-set in
X
×
X
.
Since
g
is a homeomorphism, it follows that the diagonal
Δ
X
is a
G
δ
-set
in
X
×
X
.”
Thus, the argument above proves that the next statement holds:
Propositional 3.10.
If
X
is a space with a
G
δ
-point
a
, and
g
is a
homeomorphism of
X
×
X
onto itself such that
g
(Δ
X
) =
X
× {
e
}
, for
some
e
2
X
, then the diagonal
Δ
X
is a
G
δ
-set in
X
×
X
.
Corollary 3.11.
If
X
is a rotoid such that at least one point of
X
is a
G
δ
-point in
X
, then the diagonal
Δ
X
is a
G
δ
-set in
X
×
X
.
Corollary 3.11 is more general than Proposition 5.1 in [6]. It is also
more general than Proposition 4.6 in [10].
Proof
of Theorem 3.2. By Proposition 3.9,
X
is twistable at some
e
2
X
. Since the
π
-character of
X
at
e
is countable, it follows from
Lemma 3.8 that
e
is a
G
δ
-point in
X
. Therefore, by Corollary 3.11,
X
has
a
G
δ
-diagonal in
X
×
X
. Hence,
X
is metrizable, by J. Chaber’s Theorem,
since
X
is countably compact and [14]
24
ISSN 1812-3368. Вестник МГТУ им. Н.Э. Баумана. Сер. “Естественные науки”. 2013. № 2
