is an open continuous mapping of
X
×
X
to
X
such that
g
((
x, x
)) =
y
and
g
(
Ox
×
Ox
)
Oy
. The restriction of
g
to the open subspace
Ox
×
Ox
of
X
×
X
is, clearly, the mapping
f
we are looking for.
Suppose that
X
is a topological space with a topology
T
. Then the
pointwise cardinality
of
X
at
x
2
X
is the cardinal number
min
{|
Ox
|
:
x
2
Ox
2
T
}
denoted by
|
X, x
|
. Similarly, we can define the pointwise
versions of other cardinal invariants of topological spaces. In particular,
w
(
X, x
)
will stand below for the pointwise weight of
X
at
x
2
X
.
Applying Theorem 2.9, we obtain:
Corollary 2.10.
If
X
is a diagonal-flexible space, then, for any
x, y
2
X
, we have:
|
X, x
|
=
|
X, y
|
and
w
(
X, x
) =
w
(
X, y
)
.
Recall that
the character
of a space
X
at a point
x
does not exceed
τ
(notation:
χ
(
x, X
)
≤
τ
) if there exists a base
B
x
at
x
such that
|
B
x
| ≤
τ
.
Corollary 2.11.
[6]
If
X
is a diagonal-flexible space, then, for any
x, y
2
X
, we have:
χ
(
x, X
) =
χ
(
y, X
)
.
Let
τ
be an infinite cardinal number. If
X
has a
π
-base
B
at
e
, then we
write
πχ
(
e, X
)
≤
τ
and say that the
π
-character of
X
at
e
doesn’t exceed
τ
. In this way the
π
-character
πχ
(
e, X
)
of
X
at
e
is obviously defined.
Clearly, we have:
Corollary 2.12.
If
X
is a diagonal-flexible space, then, for any
x, y
2
X
, we have:
πχ
(
x, X
) =
πχ
(
y, X
)
.
Corollary 2.13.
If
X
is a compact diagonal-flexible space, and
X
is
zero-dimensional at some point, then
X
is zero-dimensional at every point.
Theorem 2.9 and Corollaries 2.10, 2.13 and 2.12 are new results. We
will expose below a few advanced results on diagonal-flexible spaces that
have been established in [6].
Many results presented below are intimately related to the next question
which is still open:
Problem 2.14.
[6]
Is every diagonal-flexible compact space dyadic?
Note in this connection that every rectifiable compact space is a
Dugundji compactum [23] and therefore, is a dyadic compactum. See [14]
and [9] for the basic properties of dyadic compacta.
Problem 2.15.
Is it true in
ZFC
that if a sequential compact space
F
is diagonal-flexible, then
F
is metrizable?
In connection with this open problem, we present here some partial
results proved in [6]:
Theorem 2.16.
[6]
If a diagonal-flexible compact space
F
is first-
countable at some point, then
F
is metrizable.
In particular, this theorem is applicable to first-countable compacta and
to Corson compacta and Eberlein compacta [3].
Corollary 2.17.
[6]
Every diagonal-flexible Corson compactum is
metrizable.
20
ISSN 1812-3368. Вестник МГТУ им. Н.Э. Баумана. Сер. “Естественные науки”. 2013. № 2
