One of the main results below is Theorem 3.1: every compact rotoid
of countable tightness is metrizable. Using this result, we establish that if
X
is a compact rotoid such that every discrete subspace of
X
is countable,
then
X
is metrizable (Theorem 3.3). We also show that if
X
is a compact
hereditarily normal strong rotoid, then
X
is metrizable (Theorem 3.5).
These results demonstrate that the class of compact rotoids introduced
in [6] is very close to the class of dyadic compacta: practically, every
classical condition for metrizability of a dyadi compactum turns out
to guarantee metrizability of any compact rotoid as well. These results
make plausible the conjecture in [6] that every compact rotoid is a
dyadic compactum (Problem 8.19). However, whether this is so remains
unknown. I believe that the answer “yes” to the conjecture would be a
fascinating result. Of course, it would be a far reaching generalization
of the famous Ivanovskij-Kuz’minov Theorem on dyadicity of compact
topological groups (see about this [9]).
Below
τ
stands for an infinite cardinal number.
A set
A X
will be called a
G
τ
-subset of
X
, if there exists a family
γ
of open sets in
X
such that
|
γ
| ≤
τ
and
A
=
∩
γ
. If
x
2
X
and
{
x
}
is
a
G
τ
-subset of
X
, then we say that
x
is a
G
τ
-point in
X
. In this case we
also say that the pseudocharacter of
X
at
x
does not exceed
τ
, and write
ψ
(
x, X
)
≤
τ
.
The
πτ
-character of a space
X
at a point
e
2
X
is not greater than
τ
(denoted by
πτχ
(
e, X
)
≤
τ
) if there exists a family
γ
of nonempty
G
τ
-sets
in
X
such that
|
γ
| ≤
τ
and every open neighbourhood of
e
contains at least
one element of
γ
. Any such family
γ
is called
a
πτ
-network
at
e
. If
τ
=
ω
,
we rather use expressions
πω
-character and
πω
-network. In particular, if
X
has a countable
π
-base at
e
, then
πωχ
(
e, X
)
≤
ω
.
A space
X
is a rotoid if there exists
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
[6].
If
e
in this definition can be chosen to be an arbitrary element of
X
,
we say that
X
is a strong rotoid. Clearly, every homogeneous rotoid is a
strong rotoid.
Notice that every rectifiable space is a strong rotoid. Hence, every
topological group is a strong rotoid [6].
3.1. Main new results.
Here we just formulate the main results. Their
proofs are given in the next subsection.
Theorem 3.1.
Every compact rotoid of countable tightness is metrizable.
Theorem 3.2.
If
X
is a countably compact rotoid such that the
π
-
character of
X
is countable at every point of
X
, then
X
is metrizable.
In connection with the last two statements, we notice that there exists
a non-metrizable countably compact strong rotoid
X
with countable
22
ISSN 1812-3368. Вестник МГТУ им. Н.Э. Баумана. Сер. “Естественные науки”. 2013. № 2
