Under
(
CH
)
, Theorem 2.16 also covers the cases of diagonal-flexible
sequential compacta and diagonal-flexible compacta of the cardinality
≤
2
ω
. Every compact rectifiable space of countable tightness is metrizable
[15, 23]. It is not yet known whether a similar statement can be proved in
ZFC for diagonal-flexible compacta of countable tightness. However, we
have the following result:
Theorem 2.18.
[6]
If
(
CH
)
holds, then every diagonal-flexible compact
sequential space
X
is metrizable.
Proof.
Under
(
CH
)
, every nonempty sequential compactum is first-
countable at some point [8]. It remains to apply Theorem 2.16.
Consistently, every diagonal-flexible compact space of countable
tightness is metrizable [6].
Theorem 2.19.
[6]
Is every locally compact diagonal-flexible space
paracompact?
Note that every locally compact topological group is paracompact [9].
A space
X
has a
regular
G
δ
-diagonal
if there exists a countable family
of open neighbourhoods of the diagonal in
X
×
X
the intersection of the
closures of which in
X
×
X
is the diagonal
Δ
X
[18].
Proposition 2.20.
[6]
If a diagonal-flexible space
X
is first-countable
at some point, then
X
has a regular
G
δ
-diagonal.
Theorem 2.21.
[6]
If a diagonal-flexible pseudocompact space
X
has
a
G
δ
-point, then
X
is metrizable (and compact).
Proof.
Since the space
X
is pseudocompact, it is first-countable at every
G
δ
-point. By Proposition 2.20,
X
has a regular
G
δ
-diagonal. Since every
pseudocompact space with a regular
G
δ
-diagonal is metrizable [7, 18], the
space
X
is metrizable.
A powerful general metrization theorem for rectifiable spaces has been
obtained by A.S. Gul’ko [15]:
Theorem 2.22.
A nonempty rectifiable space
X
is metrizable if and
only
X
has a countable
π
-base at some point
e
2
X
.
This result doesn’t extend to all diagonal-flexible spaces, since the
Sorgenfrey line is a first-countable non-metrizable diagonal-flexible
space [10].
Problem 2.23.
Is every diagonal-flexible space with countable
π
-cha-
racter first-countable?
Problem 2.24.
Is every diagonal-flexible compact space with countable
π
-character first-countable?
3. Rotoids.
In this section, we present several new results on spaces
which are very close to diagonal-flexible spaces. They are called rotoids.
In particular, we answer a question raised in [6, Problem 8.23], and discuss
some corollaries of this result. We also formulate some open questions.
ISSN 1812-3368. Вестник МГТУ им. Н.Э. Баумана. Сер. “Естественные науки”. 2013. № 2
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