H. Bennett, D. Burke, and D. Lutzer have obtained a result of principal
importance in [10]: they have shown that not every diagonal-flexible
space is rectifiable. In fact, they established that the Sorgenfrey line is
diagonal-flexible, while it is not rectifiable, since every first-countable
rectifiable space is metrizable, by an important theorem of A.S. Gul’ko [15].
At present, we have no characterization of rectifiability or of diagonal-
flexibility of a space
X
in purely topological restrictions imposed on
X
.
This is so even if
X
is compact. it is also not clear, when a compact
subspace of a finite-dimensional Euclidean space is diagonal-flexible , and
when it is rectifiable. The question is open even for compact manifolds.
For example, the sphere
S
n
is homeomorphic to a topological group
if and only if
n
2 {
0
,
1
,
3
}
. The sphere
S
7
is rectifiable, but is not
homeomorphic to any topological group [23]. Thus, not every compact
rectifiable space is homeomorphic to some topological group.
If
n /
2 {
0
,
1
,
3
,
7
}
, then
S
n
is not rectifiable [23].
Problem 2.4.
Is every compact connected
1
-dimensional rectifiable
space metrizable?
Problem 2.5
Is every compact connected
1
-dimensional diagonal-
flexible space metrizable?
Problem 2.6
.
Is every compact connected
1
-dimensional rectifiable
space homeomorphic to
S
1
?
Problem 2.7.
Is every compact connected
1
-dimensional diagonal-
flexible metrizable space homeomorphic to
S
1
?
The next two simple facts were observed in [6]: the topological product
of any family of diagonal-flexible spaces is a diagonal-flexible space, and
every discrete space is diagonal-flexible.
However, the free topological sum of diagonal-flexible spaces needn’t
be diagonal-flexible. This is so, since the next fact holds [6]:
Proposition 2.8.
If a diagonal-flexible space
X
has an isolated point,
then
X
is discrete.
The last statement can be strongly generalized, in the following way:
Theorem 2.9.
If
X
is a diagonal-flexible space, then, for any
x, y
2
X
and any open neighbourhood
Oy
of
y
in
X
, there exists an open
neighbourhood
Ox
of
x
in
X
and an open continuous mapping
f
:
Ox
×
×
Ox
→
Oy
such that
f
((
x, x
)) =
y
.
Proof.
Since
X
is diagonal-flexible, we can fix a homeomorphism
h
of
X
×
X
onto
X
×
X
such that
h
(Δ
X
) =
X
×{
y
}
. Then
h
((
x, x
)) = (
a, y
)
,
for some
a
2
X
. Then
X
×
Oy
is an open neighbourhood of
(
a, y
)
in
X
×
X
. Since
h
is continuous, we can find an open neighbourhood
Ox
of
x
in
X
such that
h
(
Ox
×
Ox
)
X
×
Oy
. The projection
p
2
of
X
×
X
to
X
given by the formula
p
2
(
x
1
, x
2
) =
x
2
, for any
(
x
1
, x
2
)
2
X
×
X
, is an
open continuous mapping. Therefore, the composition
g
=
p
2
h
of
h
and
p
2
ISSN 1812-3368. Вестник МГТУ им. Н.Э. Баумана. Сер. “Естественные науки”. 2013. № 2
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