any
x, y
2
X
and some
z
2
X
. A thorough discussion of this notion was
given in [11, 12, 15, 23].
It is not difficult to show that every topological group is diagonal-
flexible. However, what is really amazing is that it is quite difficult to
present an example of a diagonal-flexible space which is not homeomorphic
to a topological group.
In this article, a brief survey of known results on diagonal-flexible and
rectifiable spaces is provided, a list of attractive open questions is given,
and a sample of new results on rotoids is presented, accompanied by an
extensive list of references.
In general, our terminology and notation follow [14]. Under a space we
understand a Tychonoff topological space. The
tightness
of a space
X
is
countable if the closure of any subset
A
of
X
is the union of the closures
of countable subsets of
A
. A compact space
B
is said to be
dyadic
if it can
be represented as a continuous image of a compact topological group. All
metrizable compacta are dyadic [14].
2. Some properties of rectifiable spaces and of diagonal-flexible
spaces.
The closed unit interval
I
= [0
,
1]
is not diagonal-flexible. Indeed,
the complement of the diagonal in the square
I
×
I
is disconnected, while
the complement to
I
× {
0
}
is connected. Therefore, the diagonal cannot
be mapped onto
I
× {
0
}
by a homeomorphism of the square onto itself.
Note, however, that the diagonal in
I
×
I
can be mapped onto
I
× {
1
/
2
}
by a homeomorphism of the square onto itself.
The next simple, but quite important, theorem from [11, 23] shows that
the class of diagonal-flexible spaces is very wide.
Theorem 2.1.
Every topological group
G
is a diagonal-flexible space.
Proof.
Put
h
(
x, y
) = (
x, xy
−
1
)
, for each
(
x, y
)
2
G
×
G
. Clearly,
h
is a
homeomorphism of
G
×
G
onto
G
×
G
mapping the diagonal onto
G
×{
e
}
.
Since
G
is homogeneous, we can move the diagonal by a homeomorphism
to
G
× {
z
}
for any
z
2
G
as well.
Every topological group is, in fact, a rectifiable space, as the above
argument demonstrates.
Theorem 2.1 does not generalize to paratopological groups, since the
Sorgenfrey line is a paratopological group but is not rectifiable, as was
shown in [15]. The next question is open, and it would be very nice to
learn the answer to it:
Problem 2.2.
Is every compact diagonal-flexible space rectifiable?
We see below that not every rectifiable space is homeomorphic to a
topological group (the sphere
S
7
witnesses this). However, every rectifiable
space is homogeneous [15]. The next question remains open:
Problem 2.3.
[6]
Is every diagonal-flexible space homogeneous?
18
ISSN 1812-3368. Вестник МГТУ им. Н.Э. Баумана. Сер. “Естественные науки”. 2013. № 2
