\sim
も
tr
$\mathrm{i}$zabi
1
$\mathrm{i}$
ty
of
s 暇下 Ces
h
下
vi
$\mathrm{n}\mathrm{g}$certai
$\mathrm{n}\mathrm{k}^{-}\mathrm{n}\mathrm{e}\mathrm{t}_{\mathrm{V}}0\mathrm{r}\mathrm{k}\mathrm{s}$
中祥雄
(Yoshio Tanaka)
Department
of
Mathematics,
Tokyo Gakugei University
As is
well-lmown,
each of the
following
properties
implies
that X is
metrizable.
(A)
X is
a
paracompact
developable
space
(R.
H.
Bing [2]).
(B)
Xis
aparacompact
space
having
a
$\sigma-1\propto \mathrm{a}\mathrm{l}\mathrm{l}\mathrm{y}$cour
市小
le
base
(V.
V.
Fedorcuk
[9]).
(C)
X
has
a
$\sigma$-hereditarily
cloeure-praeerving
base
(D. Burke,
R.
Engelking
and
D. Lutzer
$[5|)$
.
(D)
X is
a
paracompact
$\mathrm{M}$-space,
and
a
$\sigma-\mathrm{s}\mathrm{p}\mathrm{a}\mathrm{C}\mathrm{e}$(A.
Okuyama [15],
the
paracompactness
can
be
omitted;
see
F. Siwiec
and
J.
Nagata [16]
$)$
.
(E)
X
is
an
$\mathrm{M}$-space
having
a
point-coumtable
base
(V.
V.
Filippov [10]).
In
terms
of
these
properties,
we
give
some
metrization
theorems
by
means
of certain
$\mathrm{k}$-networks,
or
generalizations
of
M-spaces,
etc.
We
assume
that spaces
are
regul
$ar$
and
$T_{1}$
.
Def
$\mathrm{i}$ni ti
$\mathrm{o}\mathrm{n}\mathrm{s}$
.
(1)
A
cover
$C$
of
a
space
a
$k\tau wtumk$
if,
whenever
$\mathrm{K}\subset \mathrm{U}$
with
$\mathrm{K}$compact
and
$\mathrm{U}$open
in
X,
then
$\mathrm{K}\subset\cup C’\subset \mathrm{U}$
for
some
finite
$C’\subset C$
.
If the
$\mathrm{K}$is
a
single
point,
then such
a
cover
is
called network
(or
net).
Recall that
a
space is
an
$\#$
-space
(resp.
$\sigma-\mathrm{s}_{\mathrm{P}}\mathrm{a}\mathrm{C}\mathrm{e}$)
if
it
has
a
o-locally
(2)
A
space
is
courtabl $ybi-quaSi-k[14]$
if,
whenever
(
$\mathrm{F}_{\mathrm{n}}$:
$\mathrm{n}\in \mathrm{N}$
}
is
a
decreasing
sequence with
$\mathrm{x}\in\overline{\mathrm{F}}_{\mathrm{n}}$,
there
exists
a
decreasing
sequence
(
$\mathrm{A}_{\mathrm{n}}$:
$\mathrm{n}\in \mathrm{N}$
}
such that
$\mathrm{x}\in\overline{\mathrm{A}_{\mathrm{n}}\cap \mathrm{F}}_{\mathrm{n}}$for each
$\mathrm{n}\in \mathrm{N}$
,
and
if
$\mathrm{x}_{\mathrm{n}}\in \mathrm{A}_{\mathrm{n}}$for each
$\mathrm{n}\in \mathrm{N}$
,
then
$\mathrm{t}l\mathrm{E}$sequence
(
$\mathrm{x}_{\mathrm{n}}$:
$\mathrm{n}\in \mathrm{N}$
}
has
a
cluster
$\mathrm{r}\mathfrak{v}\mathrm{i}\mathrm{n}\mathrm{t}$in
$\cap(\mathrm{A}_{\mathrm{n}}$:
$\mathrm{n}\in \mathrm{N}$
}.
${\rm Re}(\mathrm{B}11$
that
a
space
X is
a
q-space
if each
point
has
a
sequence
(
$\mathrm{V}_{\mathrm{n}}$:
$\mathrm{n}\in \mathrm{N}$
}
of
nbds
such that if
$\mathrm{x}_{\mathrm{n}}\in \mathrm{V}_{\mathrm{n}}$,
the
sequence
(
$\mathrm{x}_{\mathrm{n}}$:
$\mathrm{n}\in \mathrm{N}$
}
has
a
cluster
point
in
X.
Every
$\mathrm{q}$-space is
a
countably
$\mathrm{b}\mathrm{i}^{-}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{s}\mathrm{i}-\mathrm{k}^{-\mathrm{s}}\mathrm{P}\mathrm{a}\mathrm{c}\mathrm{e}$
.
Every countably
bi-$\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{s}\mathrm{i}^{-}\mathrm{k}^{-\mathrm{s}}\mathrm{P}\mathrm{a}\mathrm{c}\mathrm{e}$is
precisely
the
countably
$\mathrm{b}\mathrm{i}$
-quotient image
of
an
M-space;
see
[14].
(3)
A
space
X
is
a
$m7\mathrm{a}\sigma mcw\Delta-sMce$
(simply,
$\mathrm{m}\mathrm{w}\Delta-\mathrm{s}\mathfrak{B}^{\mathrm{c}}\mathrm{e}$)
[18]
(raep.
$m\tau d\sigma mC$
devetoWble
[7]
(equivalently,
space
having
a
$\mathrm{l}\mathrm{H}\mathrm{s}\mathrm{e}$of countable
order in the
sense
of
Arhangel’skii
$\lfloor 1$])),
if there
exists
a
sequence
$(\mathcal{B}_{\mathrm{n}})$
of
bases for X such that
any
decreasing
sequence
(
$\mathrm{B}_{\mathrm{n}}$:
$\mathrm{n}\in \mathrm{N}$
}
with
$\mathrm{B}_{\mathrm{n}}\in\beta_{\mathrm{n}}$satisfies
$(*)$
:
$(*)$
If
$\mathrm{x}\in\cap \mathrm{B}_{\mathrm{n}}$
,
and
$\mathrm{x}_{\mathrm{n}}\in \mathrm{B}_{\mathrm{n}}$,
thaen the
sequence
$\{\mathrm{x}_{\mathrm{n}} :
\mathrm{n}\in \mathrm{N}\}$
has
a
cluster
point
in X
(resp.
the cluster
point
x).
If the
sequence
(
$\mathrm{B}_{\mathrm{n}}$:
$\mathrm{n}\in \mathrm{N}$
}
with
$\mathrm{B}_{\mathrm{n}}\in\beta_{\mathrm{n}}$is
not
necessarily decreasing,
then such
a
space
is called
a
$\mathrm{w}\Delta-\mathrm{s}\mathrm{m}\mathrm{C}\mathrm{e}$;
developable
space
respectively.
Every
$\mathrm{w}\Delta^{-\mathrm{s}_{\mathrm{P}}}\mathrm{a}\mathrm{c}\mathrm{e}$or
every
(monotonic)
$\mathrm{P}$-space[7]
is
an
$\mathrm{m}\mathrm{w}\Delta^{-_{\mathrm{S}}}\mathrm{p}\mathrm{a}\mathrm{c}\mathrm{e}$
.
Every
$\mathrm{m}\mathrm{w}\Delta$-space
is
a
$\mathrm{q}$
-space, hence
countably
$\mathrm{b}\mathrm{i}^{-\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{S}}\mathrm{i}-\mathrm{k}$
.
Among
$\theta-\mathrm{r}\mathrm{e}\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{a}\mathrm{b}\mathrm{l}\mathrm{e}(=$submetacompact)
spaces,
$\mathrm{m}\mathrm{w}\Delta^{-}\mathrm{s}_{\mathrm{P}^{\mathrm{a}\mathrm{c}\mathrm{a}\mathrm{e}}}$(resp.
monotonic
developable spaces)
are
$\mathrm{w}\Delta$-spaces
(
$\mathrm{r}\mathrm{a}\mathrm{e}_{\mathrm{I})}$
.
developable spaces);
see
[18]
(resp.
[7]).
Ihe lemma below holds
by
maens
of
[13;
$\mathrm{p}_{\Gamma \mathrm{O}\mathrm{I}\epsilon}\mathrm{x}\mathrm{i}\mathrm{t}\mathrm{i}_{0}\mathrm{n}3.2\rfloor$,
[17;
Section
4],
$[4; \mathrm{m}\infty \mathrm{r}\mathrm{e}\mathrm{m}4.1]$
,
and
[
$18\rfloor$
,
etc.
Here,
we
note
that,
in
a
space
having
a
$\sigma$
-locally
countable
$\mathrm{k}$-network,
each
point
is
a
$\mathrm{G}_{\delta}-\mathrm{S}\mathrm{e}\mathrm{t}$.
So,
every
countably
$\mathrm{b}\mathrm{i}-\eta \mathrm{u}\mathrm{a}\mathrm{s}\mathrm{i}-\mathrm{k}$-space
with
a
$\sigma$-locally
countable k-network is
countably
bi-k
by
$\mathrm{L}\sqrt$
.
(1)
$\mathrm{S}\mathrm{u}\mathrm{p}\infty \mathrm{e}$that X is
a
$\mathrm{k}$-space;
a
normal
space
in which
every
closed
countably
compact
set
is compact;
or
each
point
of
X is
a
$\mathrm{G}_{\delta}-\mathrm{S}\mathrm{e}\mathrm{t}$.
Then
(i)
and
(ii)
below hold.
(i)
X
has
a
point-countable
$1_{\mathrm{B}}\mathrm{s}\mathrm{e}$if
and
only
if X is
a
countably
bi-$\mathfrak{M}\mathrm{a}\mathrm{s}\mathrm{i}-\mathrm{k}^{-_{\mathrm{S}}}\mathrm{p}\mathrm{a}\mathrm{c}\mathrm{e}$with
a
$\iota \mathrm{r}\mathrm{i}\mathrm{n}\mathrm{t}$-countable
k-network.
(ii)
X is
a
nomotonically developable
space
with
a
point-countable
$1_{\mathrm{H}}\mathrm{s}\mathrm{e}$if
$\mathrm{a}\mathrm{I}\mathrm{H}$only
if X is
an
$\mathrm{m}\mathrm{w}\Delta$-space
with
a
$\mathrm{r}\mathrm{x}$)
$\mathrm{i}\mathrm{n}\mathrm{t}-_{\mathrm{C}}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{t}\mathrm{a}\mathrm{b}\mathrm{l}\mathrm{e}$
k-network
(cf.
[18]).
(2)
(i)
A space
has
a
$\sigma$-locally
countable
oese
if
and
only
if
it is
a
countably
$\mathrm{b}\mathrm{i}-\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{s}\mathrm{i}^{-}\mathrm{k}-\mathrm{S}\mathrm{p}\mathrm{a}\mathrm{C}\mathrm{e}$with
a
o-locally
countable k-network.
(ii)
A spaoe is
a
nomotonically developable
space with
a
$\sigma-1\propto \mathrm{a}\mathrm{l}\mathrm{l}\mathrm{y}$countable
$l_{\mathrm{H}}\mathrm{s}\mathrm{e}$if and
only
if it is
an
$\mathrm{m}\mathrm{w}\Delta-\mathrm{S}\mathrm{p}\mathrm{a}\mathrm{C}\mathrm{e}$
with
a
$\sigma-1\propto \mathrm{a}\mathrm{l}\mathrm{l}\mathrm{y}$countable k-network.
ffitri
zati
on
$\mathrm{l}\mathrm{h}\mathrm{o}\mathrm{r}\mathrm{e}\mathrm{m}$The
following
are
equivalent.
(a)
X is
metrizable,
(b)
X is
a
paracompact
$\mathrm{M}$-space
having
a
point-countable
k-network,
(c)
X is
an
M-space,
and
a
$\mathrm{k}$-space
having
a
point-countable
k-network,
(d)
X
is
an
M-space having
a
point-countable k-network,
and
having
a
$\sigma$
-locally
countable
$oetuD\gamma k$
.
(e)
X is
an
$\mathrm{M}$-space
having
a
o-locally
countable
$\mathrm{k}-\iota \mathrm{a}\mathrm{e}\mathrm{t}\mathrm{W}\mathrm{o}\mathrm{r}\mathrm{k}$
,
(f)
X is
a
paracomrct,
countably
$\mathrm{b}\mathrm{i}-\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{s}\mathrm{i}-\mathrm{k}-\mathrm{S}\mathrm{P}^{\mathrm{a}\mathrm{C}\mathrm{e}}$having
a
$\sigma-1\propto \mathrm{a}\mathrm{l}\mathrm{l}\mathrm{y}$countable
k-network,
(g)
X is
a
countably
$\mathrm{b}\mathrm{i}^{-_{\mathrm{q}\mathrm{u}\mathrm{a}}}\mathrm{s}\mathrm{i}-\mathrm{k}$-space
having
a
point-coumtable k-network,
anfi
having
a
$\sigma$-closure
preserving
k-network;
cf.
[12].
(h)
X is
an
$\mathrm{m}\mathrm{w}\Delta^{-\mathrm{S}}\mathrm{p}\mathrm{a}\mathrm{C}\mathrm{e}$having
a
$\sigma$-closure
preserving
k-network;
see
[18].
(i)
X is
a
countably
$\mathrm{b}\mathrm{i}-\mathfrak{M}^{\mathrm{a}}\mathrm{s}\mathrm{i}-\mathrm{k}$-space
having
a
$\sigma$-heraitarily
closure
kmark.
In the
previous theorem,
it
is
possible
to
replace
‘
countably
$\mathrm{b}\mathrm{i}-\tau \mathrm{u}\mathrm{a}\mathrm{S}\mathrm{i}^{-}\mathrm{k}-\mathrm{S}\mathrm{P}\mathrm{a}\mathrm{C}\mathrm{e}$”
by
.
countably
$\mathrm{b}\mathrm{i}-\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{S}\mathrm{i}-\mathrm{k}-_{\mathrm{S}}\mathrm{p}\mathrm{a}\mathrm{C}\mathrm{e}$,
but the
sequence
$\{\mathrm{x}_{\mathrm{n}} :
\mathrm{n}\in \mathrm{N}\}$
has
a
cluster
$\iota \mathrm{x}$)
$\mathrm{i}\mathrm{n}\mathrm{t}$
in X
(instead
of
$\cap(\mathrm{A}_{\mathrm{n}} :
\mathrm{n}\in \mathrm{N}\})$
in the
definition
of
countably
$\mathrm{b}\mathrm{i}^{-_{\mathrm{q}\mathrm{u}\mathrm{a}}}\mathrm{s}\mathrm{i}-\mathrm{k}$-spaces
”
We
see
that each condition in
(b)
$\sim(\mathrm{h})$
of
Etrization
Theorem is
essential
by
mfflns
of the
following examples.
Exanpl
es.
(1)
Not
every
countably
compact
(resp. countably
compact,
first
countable)
space
having
a
point-countable
$\mathrm{k}$-network
(resp. locally
countable
oetumk)
is
a
$\sigma$-space.
(2)
Not
every
Cech-CVmplete
(raep.
metacompact
developable)
space
having
a
$\sigma-1\propto \mathrm{a}\mathrm{l}\mathrm{l}\mathrm{y}$countable kse
is
a
$\sigma-\mathrm{s}\mathrm{p}\mathrm{a}\mathrm{c}\mathrm{e}[8]$(resp. metrizable;
cf.
[11]).
(3)
Not
every
first
countable, Lindel\"of
space
having
a
$\sigma$-closure
preserving
base is
developable [6].
But,
the
following
holds
by
maens
of
[3]
and
[18]
,
etc.
Roposi
ti
on
(1)
Every
$\theta$-refinable
space
X
is
developable
if
(a)
or
(b)
below holds.
(a)
X
is
an
$\mathrm{m}\mathrm{w}\Delta^{-}\mathrm{s}\mathrm{p}\mathrm{a}\mathrm{c}\mathrm{e}$having
a
$\infty \mathrm{i}\mathrm{n}\mathrm{t}^{-\mathrm{c}\mathrm{o}}\mathrm{u}\mathrm{n}\mathrm{t}\mathrm{a}\mathrm{b}\mathrm{l}\mathrm{e}\mathrm{k}$-network,
or
having
a
$\sigma$-locally
coumtable
oetuxyrk.
(b)
X
is
a
countably
$\mathrm{b}\mathrm{i}^{-\mathrm{q}\mathrm{u}8}\mathrm{S}\mathrm{i}^{-\mathrm{k}\mathrm{p}}- \mathrm{S}\mathrm{a}\mathrm{C}\mathrm{e}$having
a
$\sigma-1\mathfrak{c}\mathrm{x}\mathrm{B}\mathrm{l}\mathrm{l}\mathrm{y}$
countable
$\mathrm{k}^{-}\mathrm{I}\mathrm{E}\mathrm{t}\mathrm{W}\mathrm{o}\mathrm{r}\mathrm{k}$.
(2)
Every
$\mathrm{m}\mathrm{w}\Delta-\mathrm{S}\mathrm{p}\mathrm{a}\mathrm{C}\mathrm{e}$which is the
quotient
mmpact image
(raep. quotient
s-image)
of
a
metric
space
is
developable (resp. monotonically
developable);
We
nmte
that
every
$\mathrm{m}\mathrm{w}\Delta$-space
having
a
$\sigma$-lotBlly
finite
$mtuD\gamma k$
(resp.
$\sigma$-locally
countable
$\mathrm{k}-\mathrm{I}\mathrm{E}\mathrm{t}\mathrm{W}\mathrm{o}\mathrm{r}\mathrm{k}$
)
is
developable
(resp.
monotonically
developable;
cf.
[18].
In view
of
the
$\mathrm{a}\mathrm{l}\mathrm{p}\mathrm{V}\mathrm{e}$results,
we
have
the
following
questions.
$QStions$
.
(1)
Every
$\mathrm{w}\Delta$-space
having
a
$\sigma-1\propto \mathrm{a}\mathrm{l}\mathrm{l}\mathrm{y}$countable
k-network
(resp.
$\sigma$-locally
countable
oetumk)
is
developable
(resp.
monotonically
developable)
$\nabla$(2)
Every
$\mathrm{w}\Delta$-space which is the
quotient
$\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{t}\mathrm{a}\mathrm{b}\mathrm{l}\mathrm{e}-\mathrm{t}\mathrm{o}$
-one
image
of
a
metric
space is
developable
$\nabla$Ref
erenres
[11
A. V.
Arhangel
ski
$\mathrm{i}$,
Certain
metr
$\mathrm{i}$
zat
$\mathrm{i}$on
theorems, Usp.
Mat.
,
Nauk,
18(1963),
139-145.
[2]
R. H.
Bing,
Metr ization of
topologi
cal spaces,
Canad. J.
Math.
,
3(1951),
175-186.
[31
D.
Burke,
Refinements of
locally
countable
collections, Topology
Proceedings, 4(1979),
19-27.
[4]
.
Paralindel\"of
spaces and
closed
mappings,
ibid.
5(1980),
47-57.
[5]
D.
Burke,
R.
Engelking
and
D.
Lutzer, Hereditarily
closure-preserving
collections
and
metr
$\mathrm{i}$zat
$\mathrm{i}$on, Proc. Amer. Math.
Soc.
,
51
(1975),
483-488.
[6]
J. G.
Ceder,
Some
general
$\mathrm{i}$zat
ions of
metr
$\mathrm{i}\mathrm{c}$spaces,
Pac
$\mathrm{i}\mathrm{f}\mathrm{i}\mathrm{c}$J. Math.
,
[7]
J.
Chaber,
M.
M. Coban and K.
Nagami,
On monotonic
generalizations
of Moore
spaces,
Cech
complete
spaces and
$\mathrm{p}$-spaces,
Fund
Math.
,
84(1974),
107-119.
[81
S.
W.
Davis,
A
nondevelopable
$\mathrm{C}\mathrm{e}\mathrm{c}\nu \mathrm{h}^{-\mathrm{C}\mathrm{O}}\mathrm{m}\mathrm{P}\mathrm{l}\mathrm{e}\mathrm{t}\mathrm{e}$space with
a
point-countable
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Proc. Amer. Math. Soc.
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(1980),
139-142.
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V. V.
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Ordered
sets
and
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product
of
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(Russian),
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Mos.
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(1966),
66-71.
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V.
V.
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On feathered
paracompacta,
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,
9(1968),
161-164.
[11]
W.
G. Fleissner
and
G.
M.
Reed,
Paralindel\"of spaces
and spaces with
a
$\sigma-1_{\mathrm{o}\mathrm{C}}\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{y}$countable
base,
Topology Proceedings,
2
(1977),
89-110.
[12]
Z. M.
Gao,
The
closed
images
of
metric
spaces and
Fr\’echet
$R$
-spaces,
$\mathrm{Q}$
&
A
in
General
Topology.
,
5
(1987),
281-291.
[13]
G.
Gruenhage,
E.
Michael and
Y.
Tanaka,
Spaces
determined
by
point-countable
covers,
Pac
$\mathrm{i}\mathrm{f}\mathrm{i}\mathrm{c}$J.
Math.
,
113
(1984),
303-332.
[14]
E.
Mi
chael,
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$\mathrm{i}$ent
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Gen.
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,
2
(1971),
91-138.
[15]
A.
Okuyama,
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metr
$\mathrm{i}$zabi
1
$\mathrm{i}$ty
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176-179.
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Y.
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$\mathrm{e}\mathrm{d}\mathrm{s}$
