tightness: to see this, it is enough to take the standard
Σ
-product of
uncountably many copies of the discrete group
D
consisting of just two
elements. In particular, we see that Theorem 3.1 cannot be extended to
countably compact rotoids.
Theorem 3.3.
If
X
is a compact rotoid such that every discrete subspace
of
X
is countable, then
X
is metrizable.
Theorem 3.4.
If
X
is a compact strong rotoid such that the
π
-character
of
X
at some point of
X
is countable, then
X
is metrizable.
Recall that a space is
hereditarily normal
if every subspace of it is
normal.
Theorem 3.5.
If
X
is a compact hereditarily normal strong rotoid, then
X
is metrizable.
Theorem 3.6.
If
Z
is a retract of a strong rotoid
X
, and
Z
is a
compactum with countable tightness, then
Z
is first-countable.
Theorem 3.7.
If
Z
is a retract of a strong rotoid
X
, and
Z
is compact,
then
Z
is first-countable at every point
z
at which
Z
has a countable
π
-base.
3.2. Proofs of the main results.
B.E. Shapirovskij has shown [21] that
if
X
is any compact space of countable tightness, then the
π
-character of
X
at every point of
X
is countable. Hence, Theorem 3.1 immediately follows
from Theorem 3.2.
If every discrete subspace of a compact space
X
is countable, then
the tightness of
X
is countable as well, by a result in [1]. Therefore,
Theorem 3.3 follows from Theorem 3.1. Thus, to prove the first three
theorems, it is enough to prove Theorem 3.2.
To do this, we recall some techniques from [4].
A twister at a point
e
of a space
X
is a binary operation on
X
, written
as
xy
for
x
,
y
in
X
, satisfying the following conditions:
a)
ex
=
xe
=
x
, for each
x
2
X
;
b) for every
y
2
X
and every open neighbourhood
V
of
y
, there is an
open neighbourhood
W
of
e
such that
Wy V
; and
c) if
e
2
B
, for some
B X
, then, for every
x
2
X
,
x
2
xB
.
If a space
X
has a twister at
e
2
X
, then we say that
X
is
twistable
at
e
. A space is
twistable
if it is twistable at every point. Twistability was
introduced and applied in [4], [5]. The next fact has been established in [4].
For the sake of completeness, we provide its proof.
Lemma 3.8.
Suppose that
X
is a space twistable at a point
e
2
X
, and
that
πτχ
(
e, X
)
≤
τ
, for some cardinal number
τ
. Then
ψ
(
e, X
)
≤
τ
.
Proof.
Fix a twister at
e
, and let
γ
be a
πτ
-network at
e
. Take any
V
2
γ
, and fix
y
V
2
V
.
There exists a
G
τ
-set
P
V
in
X
such that
e
2
P
V
and
P
V
y
V
V
. Put
Q
=
∩{
P
V
:
V
2
γ
}
. Clearly,
Q
is a
G
τ
-set and
e
2
Q
.
ISSN 1812-3368. Вестник МГТУ им. Н.Э. Баумана. Сер. “Естественные науки”. 2013. № 2
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