数理解析研究所講究録
1074 巻 1999 年 32-37

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PRODUCTS OF -SPACES, AND k-NETWORKS$\mathrm{k}$中祥雄 (Yoshio Tanaka)
Department of Mathematics, Tokyo oekugei University

We give some affirmations and negations to the following $HyMfwSis$.
We assume that spaces are regular and$Hy\infty breSis$

Ihen,$\mathrm{X}\mathrm{x}\mathrm{Y}$$(\mathrm{K}_{1})\mathrm{X}$.

: Let X and be -spaces with point-countable k-networks.
is a -space if and only if one of the following holds.
$\mathrm{Y}$$\mathrm{k}$$\mathrm{k}$and(K2) X or$(\mathrm{K}_{3})\mathrm{X}$$\mathrm{T}_{1}$$\mathrm{Y}$$\mathrm{Y}$and$\mathrm{Y}$have)$\mathrm{i}\mathrm{n}\mathrm{t}$$\mathrm{I}\mathrm{x}$-countable bases.

is locally compact.

are locallyNote that theif“$\mathrm{k}_{\omega}$”$\mathrm{I}\mathrm{H}r\mathrm{t}$-spaces.

of Hyjbothesis holds for any -spaces X and Y.
$\mathrm{k}$Lasnev spaces; quotient -images of metric spaces; or$\mathrm{C}\mathrm{W}$$\mathrm{s}$-complexes are-spaces with a point-countable k-network.

$\mathrm{k}$We recall some definitions used in this note.
let$\mathrm{C}$be a cover of X.

is $wi7k$ -courtabl$\mathrm{C}$$\iota \mathrm{x})\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{x}\in \mathrm{X}$

many
$X$

$\mathrm{D}\in \mathrm{C}$isX isclosed$co77\Phi act-\tau ouTaeable;star-Cou\gamma aeable$ )(resp. compact KCX; element$\mathrm{C}\in \mathrm{C}$$\mathrm{i}\mathrm{f}$any) meets at most countably.

$de\iota_{\mathrm{e}}rmoed$for everyThen,(resp.

$\mathrm{e}$Let X be a space, and$\mathrm{C}\in \mathrm{C}$$d_{\mathrm{o}mm}ted$$\mathrm{s}\mathrm{u}1_{\mathrm{B}}\mathrm{e}\mathrm{t}$.

by$\mathrm{C}$

,

Here,
by

$\mathrm{C}$,if FCX is closed in X iff

we oeI1 replace“closedif for any subcollection

determined by$\mathrm{C}^{*}$”$\mathrm{F}\cap \mathrm{C}$by$\mathrm{C}^{*}$“ofis closed inopen$\mathrm{C}$,$\mathrm{C}$”$\mathrm{U}\mathrm{C}^{*}$is a.

A space is a $k$-stnce if it is determined by a cover of compact subsets.

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Every space is determined by any open cover.
by any HCP ($=\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}\mathrm{d}\mathrm{i}\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{i}\mathrm{l}\mathrm{y}$$\alpha$be an infinite cardinal.

$<\alpha$.

a space is aThen,of compact subsets with$C$

is local $ly<h$ if each$\alpha(\mathrm{x}\rangle$$\mathrm{x}\in \mathrm{X}$is a

A spaceX is ocal
$l$a finite ,is asuch that KC$\mathrm{D}^{*}\subset P$$c\mathrm{s}^{*}$$,\mathrm{D}$if, for any nbd$k\eta oet\mathrm{j}uMk$$\mathrm{P}$contains

A -network ,$\mathrm{k}$Obviously,-stnce if it is$|\mathrm{C}|\leqq\alpha$.

A space X$lyk_{\omega}$

if X is locally$\mathrm{U}$$<\mathrm{k}_{\alpha},$$\mathrm{c}\mathrm{s}$$\mathrm{x}$.

$\mathrm{K}$in X, there is.
,there is$\mathrm{P}\in P$

with$\mathrm{U}$of$\mathrm{x}\in \mathrm{X}$$\mathrm{x}\in \mathrm{P}\subset \mathrm{U}$,,frequently (resp. eventually).

$\mathrm{L}$$\mathrm{D}$converging to$\mathrm{L}$$\alpha=\omega_{1}$

,

Then,

of a compact set$\cup,0^{*}\subset \mathrm{U}$$\mathrm{k}_{\alpha(\mathrm{x}\rangle}-_{\mathrm{S}\mathrm{p}}\mathrm{a}\mathrm{c}\mathrm{e}$-oetuork (resp. $cs-\eta oetwrk$) if, for any open nbd

anfi for any sequence
and$k_{\mathrm{a}}$has a nbd whose closure is a

Let X be a space, and let $p$ be a cover of X.
$p$

Every CW-$\mathrm{k}$determined by a coverhereclosure-preserving) closed cover.

-space dominated by a cover of compact metric subsets.

complex is aLet

Every space is dominatedis closed if elements of $p$ are closed.

-networks;

Spaces with a countable

are $\Re_{0}-sMces$ (resp.

or closed -networks are$\mathrm{k}$$\mathrm{k}$$\mathrm{c}\mathrm{s}^{*}$-networks.

-network (resp. o-locally finite k-network)$\Re-SpaCeS$ ).

Spaces with a countable network are cosrm $csMces$ .

Letif ($\omega\omega$be the set of all functions from co to$\mathrm{n}\in\omega:\mathrm{f}(\mathrm{n})<\mathrm{g}(\mathrm{n})\}$Let BF(co 2) mean$(\mathrm{C}\mathrm{H})\Rightarrow\urcorner$

BF

Let X and
$HyWbu_{\mathrm{s}}is$$\mathrm{Y}$$(\omega_{2})$beholds$\mathrm{R}$Let$\mathrm{b}=\min${.

$|\mathrm{A}|$meanFor:$\mathrm{f},$)$\exists \mathrm{u}\mathrm{n}l\mathrm{x}$$\mathrm{g}\in^{\omega}\omega,$$\mathrm{u}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{d}$AC$\mathrm{f}\geqq \mathrm{g}$$\omega\omega$

”.

Then,

-spaces; or closed -images of metric spaces.

Then,

“,

is finite.

$\omega$$\mathrm{b}\geqq\omega_{2}$and$([\mathrm{T}_{1}])$”, and I BF$(\omega_{2})$$(\mathrm{M}\mathrm{A}+\neg \mathrm{C}\mathrm{H})\subset\Rightarrow \mathrm{B}\mathrm{F}(\omega_{2})$$\mathrm{s}$.

“$\mathrm{b}=\omega_{1}$.

}.

34

Let X andand,$\mathrm{Y}$be$\mathrm{L}\mathrm{a}^{\mathrm{v}}\mathrm{s}\mathrm{n}\mathrm{e}\mathrm{v}$spaces.

holds$\urcorner \mathrm{B}\mathrm{F}(\omega_{2})\Leftrightarrow Hywtf\mathcal{B}sis$

Let X and$\mathrm{Y}$be$\mathrm{C}\mathrm{W}$Then,$(\mathrm{C}\mathrm{H})\subset\Rightarrow Hy\ ^{\chi}fesiS$

holds$([\mathrm{T}_{2}])$.

$([\mathrm{G}])$-complexes (or closed images of CW-complexes).

$\urcorner \mathrm{B}\mathrm{F}(\omega_{2})\Leftrightarrow HyMksis$

holds

;

$([\mathrm{T}_{3}];[\mathrm{T}\mathrm{Z}])$

Ihen,

.

Now, let us consider following properties.

-compact, cosmic(C)Closed$(\mathrm{A}_{1})$Space with a$\sigma$$\sigma$$\mathrm{s}\mathrm{u}\iota_{\mathrm{B}}\mathrm{e}\mathrm{t}\mathrm{S}$-locally countableare$\mathrm{c}\mathrm{s}^{*}$(A2) Fr\’echet space with a point-countable$(\mathrm{A}_{3})$Space with a star-countable$(\mathrm{A}_{4})$Space with a)$\mathrm{I}\mathrm{x}$$\mathrm{i}\mathrm{n}\mathrm{t}$$\mathrm{o}\mathrm{e}^{*}$Space with a compact-countable$(\mathrm{B}_{1})$Fr\’echet space with a$\sigma$-network.
$\mathrm{c}\mathrm{s}^{*}$-network.

-network.

$\mathrm{r}\kappa$$\mathrm{c}\mathrm{s}^{*}$-network.

)int-countable k-network.

-HCP k-network.

$(\mathrm{B}_{3})$Space with a star-countable k-network.

$(\mathrm{B}_{4})$Space with a$(\mathrm{B}_{5})$Space with a compact-countable k-network.

Then,$(\mathrm{A}_{\mathrm{i}})$or$\sigma-\mathrm{c}\mathrm{o}\mathrm{m}_{\mathrm{P}\mathrm{a}}\mathrm{C}\mathrm{t}$$(\mathrm{B}_{\mathrm{i}})\Rightarrow(\mathrm{C})$-finite k-network.

;

(A2)

$\Leftrightarrow(\mathrm{B}_{1})$We note that first countable spaces withLet X lae a space, and letsequences in X with$\mathrm{L}_{\gamma}.arrow \mathrm{x}_{\gamma}\cdot\not\in \mathrm{L}_{7}$$\mathrm{t}\mathrm{L}_{\mathit{7}^{*}}:$$(\mathrm{B}_{4})$$(\mathrm{B}_{3})\mathrm{r}\Rightarrow(\mathrm{B}_{4})$,etc.

are metric, etc.

} be a collection of disjoint:} , hereLet.

$\mathrm{S}=\mathrm{U}\mathrm{t}\mathrm{L}_{\tau}$$\tau<\omega_{1}$$\mathrm{U}\mathrm{L}$$\mathrm{L}=$, let. For a compact metric subset of X with, where}.
be a subspace of X withby identifying all
is a quotient space obtained fromis the topological sum ofof to a single ) , and$\mathrm{K}\cap \mathrm{S}=\mathrm{L}$$\mathrm{K}$$\mathrm{K}^{*}/\mathrm{K}=(\Sigma \mathrm{L}_{\mathrm{Y}^{*}})/\mathrm{L}$$\mathrm{K}^{*}=\mathrm{S}\cup \mathrm{K}$$\mathrm{K}^{*}$$\mathrm{K}^{*}/\mathrm{K}$$\iota \mathrm{x}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{s}$(B2) or;$\mathrm{t}\mathrm{L}_{r}.:_{\mathrm{Y}}<\omega_{1}$$\mathrm{t}_{\mathrm{X}_{r}:\mathrm{v}}.<\omega_{1}\}$Here,-spaces.

-countable cs-network.

$(\mathrm{A}_{5})$(B2) Space with a$*_{0}$$\mathrm{i}\mathrm{n}\mathrm{t}$$\mathrm{K}$$\gamma^{-}<\omega_{1}\}$$\iota \mathrm{x}$.

$\Sigma \mathrm{L}_{\mathcal{T}^{*}}$$\mathrm{L}_{7^{*}}=\mathrm{L}_{\gamma^{\cup i\mathrm{x}}}\tau$

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By

$\underline{\mathrm{t}\mathrm{h}\mathrm{e}}\mathrm{X}$the X andand$\mathrm{Y}$$\mathrm{Y}$we mean the spaces X and,in HytxXusis; that is,$\mathrm{Y}$are -spaces with point-countable k-networks.
$\mathrm{k}$In the following theorem, (C) is essential by Theorem 3(3) below.

$\rceil[] \mathrm{a}\mathrm{e}\mathrm{o}\mathrm{r}\mathrm{e}\mathrm{m}1$. Let the X and(1) If neither X nor$\mathrm{Y}$satisfy (C).

$\mathrm{Y}$Then,contains a closed copy of any$HyWtk_{S}is$$\mathrm{K}^{*},$

isvalid.
(2) Hytd&sis is valid iff(3) IfIoll$Hy\iota dksis$$\mathrm{X}=\mathrm{Y},$.

is valid.

. Let the X satisfy one of$(\mathrm{A}_{\mathrm{i}})$$\mathrm{a}\mathrm{r}\mathrm{y}$and let the,$\mathrm{Y}$do so.

Then,is valid.

$HyWtr_{\mathcal{B}}SiS$[Ibe result for$[\mathrm{L}\mathrm{i}\mathrm{L}]$$\urcorner \mathrm{B}\mathrm{F}(\omega_{2})$resp.

$(\mathrm{A}_{1})$; (A2); andThe result for$(\mathrm{A}_{5})$is due to [S]; [ or$(\mathrm{A}_{4})$$\mathrm{S}$$\mathrm{L}\mathrm{i}$]; andis due to [LLi] , where spaces havecompact-countable $cl$ osed k-networks]. Let the X satisfy one ofMoll
ffien,$HyW\iota ksiS$is valid iff[The result for (B2);$\mathfrak{a}\mathrm{r}\mathrm{o}|$I. Let X and$(\mathrm{B}_{3})$$\mathrm{Y}$$\mathrm{a}\mathrm{r}\mathrm{y}$networks such that so$\mathrm{X}=\mathrm{Y},$;$\neg \mathrm{B}\mathrm{F}(\omega_{2})$$(\mathrm{B}_{5})$Then,one of$(\mathrm{K}_{3^{*}})$$(\mathrm{A}_{\mathrm{i}})$$(\mathrm{Y})\leqq 2$$Hy\iota au_{S}is$$\mathrm{i}\mathrm{n}\mathrm{t}$,$[\mathrm{L}\mathrm{T}_{1}|, [\mathrm{L}\mathrm{T}_{2}]$resp. ]

-countable closed$\mathrm{k}-$here so (X) is the sequential

is val id ff$(\mathrm{C}\mathrm{H})$$\mathrm{i}$$\urcorner$

BF$(\omega_{2})$

.

]

(MA).

Let the X satisfy one ofor.

$(\mathrm{B}_{\mathrm{i}})$do so.

is valid.

$Hy\mathrm{A}Xf_{\mathcal{B}S}is$$2$)$\iota \mathrm{x}$$\mathrm{Y}$Hytaksis is valid.

$\mathrm{X}=\mathrm{Y},$is due to [L];and so[me result is due to [S] under$\mathrm{T}\mathrm{h}_{\mathrm{o}\mathrm{r}}\mathrm{e}\mathrm{m}$If$\mathrm{k}$,and and let the,.

be -spaces with$(\mathrm{X})\leqq 2$

order of X (see [AF]).
If$(\mathrm{B}_{\mathrm{i}})$$\mathrm{a}\mathrm{r}\mathrm{y}$: One of X and$(\mathrm{B}_{\mathrm{i}})$,and let theThen, Hy&X’esis is valid, but replace$\mathrm{Y}$is locally$\mathrm{k}_{\omega}$,

and another is locally$\mathrm{Y}$satisfy$(\mathrm{K}_{3})$$<\mathrm{k}_{\mathrm{c}},$by$\mathrm{c}=2^{\omega}$.

36

Let us say that an operation isoperators

maps“$\underline{\mathrm{c}}1\mathrm{o}\mathrm{e}\mathrm{a}\mathrm{e}$

$n$

and

$\mathrm{C}\mathrm{D}$if it is the“$\underline{\mathrm{d}}\mathrm{o}\mathrm{m}\mathrm{i}\mathrm{n}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{s}$’.

$\mathrm{d}\mathrm{e}\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{K}$Let

class of all spaces obtained from Ioetric spaces underexample, spacae dominated by La\v{s}nev spacae belong toNote that spaces inspaces inCarol 1. Let X and$\mathrm{a}\mathrm{r}\mathrm{y}$and(k2) X or$(\mathrm{k}_{3})\mathrm{X}$$\mathrm{B}\mathrm{F}(\omega_{2})$$\mathrm{Y}$(metric) have$\sigma-\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{P}^{\mathrm{a}\mathrm{C}\mathrm{t}}$,$\mathrm{Y}$be spaces in the classXx(metric).
$\mathrm{k}$$\mathrm{C}\mathrm{D}$(metric).

Ihen,

is a -space iff one of the following holds.

$\mathrm{Y}$$\mathrm{k}$is locally compact metric.

$\mathrm{Y}$is dominated by a countable cover of locally compact metricIf$\mathrm{X}=\mathrm{Y}$,the result holds without I

One of X and$\mathrm{Y}$$(\mathrm{k}_{3})$$\mathrm{B}\mathrm{F}(\omega_{2})$.

byis dominated by a countable cover of locallycompact metric spaces, and another is the(3)Then, for-finite -networks, and(2) Under (MA), the result in (1) holds, but replacehere(metric) be theD.

$\mathrm{C}\mathrm{D}$of

are metric.

spacae, and so is Y.

$(\mathrm{k}_{3^{*}})$$\mathrm{C}$$\mathrm{S}\mathrm{i}\mathrm{t}\mathrm{i}_{\mathrm{o}\mathrm{n}}$(separable metric) have star-countable k-networks.

$C\mathrm{D}$(1) Under 1$(\mathrm{k}_{1})\mathrm{X}$$\mathrm{C}\mathrm{D}$$C\mathrm{D}$)$\mathrm{t}_{\mathrm{o}\mathrm{I}\mathrm{X})}\mathrm{l}\mathrm{o}\mathrm{g}\mathrm{i}\mathrm{c}\mathrm{a}1$

sum of$\mathrm{k}_{\alpha}$-spaces,.
is a -space iff X is metric.

$\alpha<2^{\omega}$$\mathrm{X}^{\omega}$$\mathrm{T}\mathrm{t}_{\mathrm{o}\mathrm{r}}\mathrm{e}\mathrm{m}$$\mathrm{k}$$3$(1) Under

complexaeX and(2) Under$\mathrm{Y}$$\mathrm{B}\mathrm{F}(\omega_{2}),$$Hy\iota\chi \mathfrak{X}heSis$

is not valid for$\mathrm{L}\mathrm{a}^{\mathrm{v}}\mathrm{s}\mathrm{n}\mathrm{e}\mathrm{v}$,CW-with star-countable k-networks.

$\mathrm{B}\mathrm{F}(\omega_{2}),$$Hy\beta\alpha u_{\mathrm{s}}iS$

is not valid for k-spaces X andpoint-countable closed k-networks (indeed, X and$\mathrm{Y}$$\mathrm{Y}$withare quotient, finite-

to-one images of locally compact metric spaces).
(3) Under
is not valid for -compact, -spaces X and,, where X is awhich is neither locally compact normetric, and has a point-countable closed -network with so, but
and are quotient -images of locally compact
is not locally$([\mathrm{L}\mathrm{i}\mathrm{L}])$$(\mathrm{C}\mathrm{H}),$$HyW\mathcal{E}m_{S}is$$\sigma$$\mathrm{k}$$\mathrm{L}\mathrm{a}\mathrm{s}\mathrm{n}\mathrm{e}\mathrm{V}\vee$$\mathrm{Y}$$\Re_{0^{-}\mathrm{S}\mathrm{P}}\mathrm{a}\mathrm{c}\mathrm{e}$$\mathrm{Y}$$\mathrm{Y}$$\mathrm{k}$$\mathrm{k}_{\omega}(\mathrm{X}$$\mathrm{Y}$$\mathrm{s}$metric sPacae) ([S] ).

$(\mathrm{Y})=3$

37. Let X and$Q\ovalbox{\tt\small REJECT} StiUn$(1) If$\mathrm{X}^{2}$

be k-spaces with point-countable k-networks.

$\mathrm{Y}$is a -space, then does X satisfy$\mathrm{k}$(2) As a characterization fortypes of properties on X,$\mathrm{Y}$$\mathrm{X}\cross \mathrm{Y}$from$(\mathrm{K}_{1})$,or$(\mathrm{K}_{3})\nabla$

to be a k-space, are there different$(\mathrm{K}_{1})$

,

(K2),

$(\mathrm{K}_{3})$,and$(\mathrm{K}_{3^{*}})$?

Ref erences[AF]

A. V. Arhanagel skiand S. P. Franklin, Ordinal invar iants for$\mathrm{i}$topological spaces, Michigan Math. , Journal, 15 (1968),[G]

G. Gruenhage,$\mathrm{k}$313-320.

-spaces and products of closed images of metricspaces, Proc. Amer. Math. Soc. , 80 (1980), 478-482.
$[\mathrm{L}\mathrm{i}\mathrm{L}]$S. Lin and C. Liu, On spaces with point-countable cs-networks,

74 (1996), 51-60.

Topology and its Appl. ,[L]C. Liu, Spaces$\mathrm{w}\mathrm{i}$Topology Proc. ,th a$\sigma$-heredi tar ly closure-preserving k-network,$\mathrm{i}$18 (1993), 179-188.

[LLi] C. Liu and S. Lin,$\mathrm{k}$-space property of product spaces, Acta. Math.

Sinica. 13(1997), 537-544.
$[\mathrm{L}\mathrm{T}_{1}]$C. Liu and Y. Tanaka, Spaces with a star-countable k-network, andrelated results, Topology and its Appl. , 74 (1996), 25-38.

$[\mathrm{L}\mathrm{T}_{2}]$C. Liu and Y. Tanaka, Star-countable k-networks, compact countable$\mathrm{k}$[Sl-networks, and relared results,$\mathrm{a}\mathrm{P}\mathrm{P}\mathrm{e}\mathrm{a}\Gamma$in Houston J. Math.

A. Shibakov, Sequentiality of products of spaces with pointcountable$[\mathrm{T}_{1}]$to$\mathrm{k}$-networks, Topology Proc. ,20 (1995), 251-270.

Y. Tanaka, A character ization for the products of$\mathrm{k}^{-}\mathrm{a}\mathrm{n}\mathrm{d}-\mathrm{s}0$-spaces

and related results, Proc. Amer. Math. Soc. , 59 (1976), 149-156.
[T2]

Y. Tanaka, A characterization for the product of closed images ofmetric spaces to be a$\mathrm{k}$-space, ibid. , 74 (1979), 166-170.

. 86

503-507.

$[\mathrm{T}_{3}]$Y. Tanaka, Products of CW-complexes, ibid.

[TZ]

Y. Tanaka and Zhou Hao-xuan, Products of closed images of CWcomplexes and$\mathrm{k}$(1987),-spaces, Proc. Amer. Math. Soc. , 92 (1984), 465-469.