数理解析研究所講究録
901 巻 1995 年 1-7

1On$\mathrm{k}\beta-\mathrm{S}_{\mathrm{P}}\mathrm{a}\mathrm{c}\mathrm{e}\mathrm{S}$and Some Other

Generalized Hetric Spaces.

小竹義朗(YoshiroKOTAKE)

Department of Mathematics,

Facul ty of Education

Gunma University

1. Introduction
In [11] Wu Lishing introduced the notion ofgeneralizes$\mathrm{k}^{-_{\mathrm{S}\mathrm{e}\mathrm{m}}}\mathrm{i}$$\mathrm{k}\beta$-spaces, which-stratifiable spaces due to Lutzer [7]. Recently

Xia Shengxiang studied the conditions under which$\mathrm{k}\beta$-spaces to be. We investigate further properties of$\mathrm{k}-\mathrm{s}\mathrm{e}\mathrm{m}\mathrm{i}-\mathrm{S}\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{f}\mathrm{i}\mathrm{a}\mathrm{b}\mathrm{l}\mathrm{e}$most of which are concerned with the metrization of
the class of$\mathrm{k}-\mathrm{s}\mathrm{e}\mathrm{m}\mathrm{i}$$\mathrm{k}\beta$$\mathrm{k}\beta$-spaces,-spaces. Since-stratifiable spaces is closely related to that ofNagata spaces, we also investigate the relationship between$\mathrm{k}\beta$-spaces and Nagata spaces.

Let (X,$\tau$such that$\tau$) be$\mathrm{x}\in \mathrm{g}(\mathrm{n}+1, \mathrm{X})\subset \mathrm{g}(\mathrm{n}, \mathrm{x})$Such a functionfunction).

a space and let$\mathrm{g}$$\mathrm{g}$be a function fromfor eachin X and$\mathrm{x}$is called a COC-function ($=$$\mathrm{n}$$\mathrm{N}\cross \mathrm{X}$intoin N.

countable open covering

In [31 Hodel introduced some important generalized metricspaces by means of a function

COC-function$\mathrm{g}:\mathrm{N}\cross \mathrm{X}arrow\tau$.

For the definitions of some generalized metric spaces which are

not defined in this note, see[1],[21,and [3].

Unless otherwise stated, all topological spaces are assumed tobe$\mathrm{T}_{1}$. The set of positive integers will be denoted by N.

2

2. Nagata spaces and$\mathrm{k}\beta$-spaces.

Instead of giving the original definitions of$\mathrm{k}-\mathrm{s}\mathrm{e}\mathrm{m}\mathrm{i}$spaces [71 and Nagata spaces, we present an equivalent-stratifiable

fomulations

which are used in this paper. For the actual definitions of theseconcepts, the reader is referred to [3] and [7].

Definition 2. 1 ([11], [12]). (a): A space X is a$\mathrm{k}-\mathrm{s}\mathrm{e}\mathrm{m}\mathrm{i}$such that-stratifiable space if there is a COC-function(where,is compact ) then thefor$\mathrm{g}$$(\mathrm{n}, \mathrm{X}_{\mathrm{n}})$$\mathrm{g}$sequence$\mathrm{n}=1,2$$\cap \mathrm{K}\neq\phi$the sequencea$\cap \mathrm{K}\neq\phi$$\mathrm{g}(\mathrm{n}, \mathrm{x}_{\mathrm{n}})$.

$\mathrm{K}$has a cluster point in K.

$\langle \mathrm{x}_{\mathrm{n}} \rangle$(b): A space X isthat$\cdots$$\langle \mathrm{x}_{\mathrm{n}} \rangle$$\mathrm{k}\beta$-space if there is a COC-function

for$\mathrm{n}=1,2$,$\cdots$.

(where$\mathrm{g}$suchis compact ) then$\mathrm{K}$has a cluster point.

Definition 2. 2 ([31). (a): A space X is a Nagata space if there,,forsuch thatis a COC-function$\mathrm{g}(\mathrm{n}, \mathrm{p})\mathrm{n}\mathrm{g}(\mathrm{n}, \mathrm{x}_{\mathrm{n}} )$$\mathrm{n}=1,2$$\neq\emptyset$$\cdots$$\mathrm{g}$then$\mathrm{p}$is a cluster point of the sequence(b):thatA space X is a$\mathrm{g}(\mathrm{n}, \mathrm{p})\cap \mathrm{g}(\mathrm{n}, \mathrm{x}_{\mathrm{n}} )$$\mathrm{w}\mathrm{N}$$\langle \mathrm{x}_{\mathrm{n}} \rangle$-space if there is a COC-function$\neq\emptyset$for$\mathrm{n}=1,2$,$\cdots$. then the sequence$\mathrm{g}$such$\langle \mathrm{x}_{\mathrm{n}} \rangle$

has a cluster point.

Theorem 2. 3.
A space (X,Every$\tau$)$\mathrm{w}\mathrm{N}$-space is a$\mathrm{k}\beta$-space.

is called weakly subsequential if each sequence in

X which has a cluster point has a subsequence with compact closure.
such that-space if there is a COC-function
A spaceX is a$\mathrm{g}$$\mathrm{w}\sigma$$\mathrm{p}\in \mathrm{g}(\mathrm{n}, \mathrm{y}_{\mathrm{n}} )$,$\mathrm{y}_{\mathrm{n}}\in \mathrm{g}(\mathrm{n}, \mathrm{x}_{\mathrm{n}})$for$\mathrm{n}=1,2,$

has a cluster point, see [2] and [4].

$\cdots$. then the sequence$\langle \mathrm{x}_{\mathrm{n}} \rangle$

3Theorem 2. 4.
$\mathrm{w}\sigma$

Every weakly subsequential$\mathrm{k}\beta$-space is a-space.

A space X is$\mathrm{c}$-stratifiable if there is a COC-functionthat for each comPact set

satisfies$\mathrm{p}\not\in \mathrm{C}1(\mathrm{g}(\mathrm{n}, \mathrm{K}))$in X and$\mathrm{K}$$\mathrm{p}\in \mathrm{X}-\mathrm{K}$such$\mathrm{g}$then there exists,. A space X is called$\mathrm{c}$which$\mathrm{n}$-Nagata space if it is-stratifiable and first countable.

$\mathrm{c}$A space X is a Nagata space if and only if X is aTheorem 2. 5$\mathrm{c}$-Nagata,$\mathrm{k}\beta$

Proof.
$\mathrm{h}:\mathrm{N}\cross \mathrm{X}arrow\tau$-space.

Let$\mathrm{f}$be a$\mathrm{c}$-Nagata function andbe defined by$\mathrm{h}(\mathrm{n}, \mathrm{x})=\mathrm{f}(\mathrm{n}, \mathrm{x})\cap \mathrm{g}(\mathrm{n}, \mathrm{X})$$\mathrm{k}-\mathrm{s}\mathrm{e}\mathrm{m}\mathrm{i}$that X is$\mathrm{k}^{-_{\mathrm{S}\mathrm{e}\mathrm{m}}}\mathrm{i}-\mathrm{S}\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{f}\mathrm{i}\mathrm{a}\mathrm{b}\mathrm{l}\mathrm{e}$$\mathrm{h}(\mathrm{n}, \mathrm{x}_{\mathrm{n}} )$$\cap \mathrm{K}\neq\emptyset\ni \mathrm{y}_{\mathrm{n}}$has a cluster point

for

is an

$\mathrm{m}$. Let$\mathrm{n}=1,2$$\langle \mathrm{x}_{\mathrm{n}\mathrm{k}}\rangle$

has a cluster pointsuch that Clf$\mathrm{i}$be a compact subset in X and$\mathrm{K}$,$\cdots$$(\mathrm{m}, \mathrm{C})\geq \mathrm{q}$such that$\mathrm{y}_{\mathrm{n}\mathrm{i}}\not\in \mathrm{h}(\mathrm{m}, \mathrm{X}_{\mathrm{n}\mathrm{i}})\supset \mathrm{h}(\mathrm{n}_{\mathrm{i}} .

$\mathrm{q}$$\mathrm{K}\cap \mathrm{C}=\phi$. Since$\mathrm{V}\ni \mathrm{y}_{\mathrm{n}\mathrm{i}}$\mathrm{x}_{\mathrm{n}\mathrm{i}})$-Nagata,$\mathrm{w}\mathrm{N}$$\mathrm{f}$. Then Clh.

so thatCorollary 2. 6 (Lee [5]).
$\mathrm{c}$$\mathrm{k}\beta$-space,$\langle \mathrm{x}_{\mathrm{n}} \rangle$

which converges to P.

contradiction. It follows that X is aonly if X is a. As X is a. Suppose that

we can assume thatthen there is an. Since every first. Since every -Nagata space is first countable,$\{\mathrm{p}\}\cup\{\mathrm{x}_{\mathrm{n}\mathrm{k}}|\mathrm{k}=1,2, \cdots\}=\mathrm{C}$$\langle \mathrm{y}_{\mathrm{n}\mathrm{l}\mathrm{c}}\rangle$-function. Let$\mathrm{c}$$\mathrm{p}$there is a subsequence
generality,$\mathrm{k}\beta$-stratifiable is a Nagata space, it suffices to showcountable

Let

be a

$\mathrm{g}$$\mathrm{P}\not\in \mathrm{K}$. Without loss of. Since

is a$\mathrm{c}$$\geqq \mathrm{m}$. Let$\mathrm{k}=1,2$$\mathrm{V}=\mathrm{X}$-Clh$(\mathrm{m}, \mathrm{C})$\mathrm{x}_{\mathrm{n}\mathrm{i}})$-stratifiable space.

A space X is a Nagata space if and-space.

A space X is said to have a,. So we have
. This is a$\mathrm{y}_{\mathrm{n}\mathrm{i}}\not\in \mathrm{h}(\mathrm{n}_{\mathrm{i}} .

$\mathrm{k}-\mathrm{s}\mathrm{e}\mathrm{m}\mathrm{i}$,$\mathrm{G}_{\theta^{*}}$$\cdots$,-Nagata function, there$(\mathrm{m}, \mathrm{C})\not\supset \mathrm{q}$$\mathrm{n}_{\mathrm{i}}$$\mathrm{y}_{\mathrm{n}\mathrm{k}}\in \mathrm{K}$-diagonal if there exists a,

4sequence$\langle F_{\mathrm{n}} \rangle$of open covers of X such that, for each$\bigcap_{\mathrm{n}=1}^{\infty}\mathrm{C}\mathrm{l}(\mathrm{s}\mathrm{t}(\mathrm{x}, d_{\mathrm{n}}^{*}))=\{\mathrm{x}\}$Theorem 2. 7.
X is a$\mathrm{q}$,$\mathrm{k}\beta$

A regular space X is a Nagata space if and only if-space with ais first countable, letLet$\mathrm{g}:\mathrm{N}\cross \mathrm{X}arrow\tau$$\mathrm{G}_{\delta^{*}}$-diagonal.

be a$\mathrm{h}(\mathrm{n}, \mathrm{p})\cap \mathrm{h}(\mathrm{n}, \mathrm{X}_{\mathrm{R}} )$such that$\mathrm{q}$-space in which points are$\mathrm{k}\beta$-function. Let. To show that$\neq\phi$,for$\mathrm{n}=1,2,$$\cdots$$\mathrm{h}(\mathrm{n}, \mathrm{p})\cap \mathrm{h}(\mathrm{n}, \mathrm{x}_{\mathrm{n}} )$$\mathrm{h}:\mathrm{N}\cross \mathrm{X}arrow\tau$is a$\mathrm{h}$,for$\mathrm{n}=1,2,$$\cdots$$\mathrm{w}\mathrm{N}-\mathrm{f}\mathrm{u}\mathrm{n}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}$. Then$\langle \mathrm{x}_{\mathrm{n}}\rangle$then$\mathrm{K}$$\mathrm{w}\mathrm{N}$$\mathrm{w}\mathrm{N}$be defined by-function,$\ni \mathrm{y}_{\mathrm{n}}$for all$\langle \mathrm{y}_{\mathrm{n}} \rangle$

letis compact and-space with a$\mathrm{n}\in \mathrm{N}$. Sinceconverges to

has a cluster point. So. Note that a regular-sets. Then there is a sequencea first countable function, the sequence$\mathrm{K}=\{\mathrm{p}\}\cup\{\mathrm{y}_{\mathrm{n}} | \mathrm{n}=1,2, \cdots\}$$\mathrm{G}_{\theta}$

be a first countable function.

$\mathrm{f}:\mathrm{X}\cross \mathrm{N}arrow\tau$$\mathrm{h}(\mathrm{n}, \mathrm{x})=\mathrm{f}(\mathrm{n}, \mathrm{x})\cap \mathrm{g}(\mathrm{n}, \mathrm{x})$$\langle \mathrm{y}_{\mathrm{n}}\rangle$,

see [3].

,Since every regular

Proof.

$\mathrm{x}\in \mathrm{X}$$\mathrm{G}_{\theta^{*}}$$\mathrm{g}(\mathrm{n}, \mathrm{x}_{\mathrm{n}})$$\mathrm{h}$$\mathrm{f}$$\mathrm{p}$is. Let$\cap \mathrm{K}\neq\phi$is a-diagonal is a

Nagata space (see, Kotake [5]), whence X is a Nagata space.

A space X is said to have a regular$\Delta$-diagonal if the diagonal

is the intersection of countably many closures of open subsets of$\mathrm{X}\cross \mathrm{X}$(see [5]).

Theorem 2. 8.
$\mathrm{G}_{\delta}$$\mathrm{G}_{\mathit{8}}$-diagonal is aEvery regular$\mathrm{k}-\mathrm{s}\mathrm{e}\mathrm{m}\mathrm{i}$$\mathrm{k}\beta-\mathrm{s}\mathrm{p}\mathrm{a}\mathbb{C}\mathrm{e}$-stratifiablewith a regular$\mathrm{s}_{\mathrm{P}^{\mathrm{a}\mathrm{C}\mathrm{e}}}$.

53.

letrizability ofTheorem 3. 1.
Hausdorff

7,$\mathrm{h}:\mathrm{N}\cross \mathrm{X}arrow\tau$a$\mathrm{k}-\mathrm{s}\mathrm{e}\mathrm{m}\mathrm{i}$-spaces-

A space X is metrizable if and only if X is a$\mathrm{k}\beta$

LetProof.

$\mathrm{k}\beta$-space.

be a 7-function and$\mathrm{f}$be defined by-function. Let$\mathrm{h}(\mathrm{n}, \mathrm{x})=\mathrm{f}(\mathrm{n}, \mathrm{x})\mathrm{n}\mathrm{g}(\mathrm{n}, \mathrm{x})$. To show that-stratifiable function, let$\mathrm{h}(\mathrm{n}, \mathrm{x}_{\mathrm{n}} )$$\cap \mathrm{K}\neq\emptyset$

sequencefor,$\langle \mathrm{x}_{\mathrm{n}} \rangle$

be a$\mathrm{k}\beta$$\mathrm{n}=1,2,$$\cdots$

is a$\mathrm{g}$has a cluster point$\mathrm{p}$$\mathrm{k}\beta$$\mathrm{p}$. Let$\mathrm{C}=\{\mathrm{p}\}\cup\{\mathrm{x}_{\mathrm{n}\mathrm{k}}|\mathrm{k}=1,2, \cdots\}$loss of generality thatis an$\mathrm{g}$no(no , C)

such that$\supset \mathrm{g}$(no ,$\mathrm{g}$$\mathrm{x}_{\mathrm{n}\mathrm{k}}$$\mathrm{C}\cap \mathrm{K}=\emptyset$(no , C)). If[7] ,$)$,$\mathrm{f}$7-space,X isthat converges

we may assume without
is a 7-function, there,. Now for, so$\cap \mathrm{K}=\phi$A contradiction. It follows that X is$\mathrm{k}-\mathrm{s}\mathrm{e}\mathrm{m}\mathrm{i}-\mathrm{S}\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{f}\mathrm{i}\mathrm{a}\mathrm{b}\mathrm{l}\mathrm{e}$$\langle \mathrm{x}_{\mathrm{n}\mathrm{k}}\rangle$$\mathrm{p}\not\in \mathrm{K}$. Since$\supset \mathrm{g}(\mathrm{n}_{\mathrm{k}} , \mathrm{x}_{\mathrm{n}\mathrm{k}})$-function, the. Since X is a

first countable. Then there is a subsequence

tois$\mathrm{h}$be a compact subset of X and let$\mathrm{K}$. As$\mathrm{g}$$\mathrm{n}_{\mathrm{k}}\geqq$

no$\mathrm{h}(\mathrm{n}_{\mathrm{k}} , \mathrm{x}_{\mathrm{n}\mathrm{k}})\cap \mathrm{K}=\phi$$\mathrm{k}-\mathrm{s}\mathrm{e}\mathrm{m}\mathrm{i}-\mathrm{S}\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{f}\mathrm{i}\mathrm{a}\mathrm{b}\mathrm{l}\mathrm{e}$.

. Since every

first countable space is a Nagata space (Lutzer

X is paracompact. In [3], Hode1 proved that every$\beta$,7-spaceis developable. It is well known that every paracompact developable
space metrizable, it follows that X is metrizable.

Theorem 3. 2.

is a$\mathrm{w}\theta$.

$\mathrm{k}\beta$

A regular spaceX is metrizable if and only if X-spacewith a$\mathrm{G}_{\theta^{*}}$-diagonal.

Corollary 3. 3 (Hodel [3]). A regular space X is metrizable

if and only if X is a

In [3]$\mathrm{w}\theta$,$\mathrm{w}\mathrm{N}$-spacewith a$\mathrm{G}_{\mathit{8}^{*}}$-diagonal.

Hodel noted that every developable space is a$\mathrm{w}\theta$-space.

6Therefore. we have the following corollary.
Corollary 3. 4.

A Hausdorff developable,

Corollary 3. 5 (Hodel [3]).
$\mathrm{w}\mathrm{N}$$\mathrm{k}\beta$-space is metrizable

Every Hausdorff developable,-space is metrizable.

Since every$\mathrm{k}-\mathrm{s}\mathrm{e}\mathrm{m}\mathrm{i}$-stratifiable space has a$\mathrm{G}_{\delta^{*}}$-diagonal,

Theorem 3. 2 generalizes the following result of Martin.
A regular space X is metrizable if

Corollary 3. 6 (Martin [10]).

and only if X is a$\mathrm{k}^{-_{\mathrm{S}\mathrm{e}\mathrm{m}}}\mathrm{i}-\mathrm{S}\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{f}\mathrm{i}\mathrm{a}\mathrm{b}\mathrm{l}\mathrm{e}$,$\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{s}\mathrm{i}-7^{-}\mathrm{s}\mathrm{p}\mathrm{a}\mathrm{C}\mathrm{e}$.

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