数理解析研究所講究録
1531 巻 2007 年 27-30

27

On existence of continuous selection for flnitesets
愛馬六掌理工学研窮科塾楠 (Nan$\mathrm{J}\mathrm{i}\mathrm{a}_{\mathfrak{B}}$)

Graduate School of Science and Engineering
Ehime University
Abstract, toWe prove a theorenl which allows, under suitable.
extend a Vietoris continuous selection for two-point subsets of adorfr space $X$ to a Vietoris continuous selection on all flnite subsets
of $X$ In particular, such an extension is always possible if $X$ has atpoint. We then apply this extension theorem
most oneto obtain theresult: If is a scattered hereditarily paracompact lIausdorff space wbch has a Vietoris continuous selection for
two-point subsets of $X,$ tben $X$ also has a Vietoris continuous selection
br all ffiite subsets of $X$ , This gives a partial answer to a question ofGutev and Nogura [5].
$\infty \mathrm{n}\mathrm{d}\mathrm{i}\mathrm{t}\dot{\mathrm{m}}\mathrm{n}\epsilon$$\mathrm{H}\mathrm{a}\mathrm{u}\epsilon$.

$\mathrm{n}\mathrm{o}\mathrm{n}\cdot \mathrm{i}\mathrm{s}\mathrm{o}\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{d}$$X$$\mathrm{f}\mathrm{o}\mathbb{I}\mathrm{o}\mathrm{w}_{\grave{\mathfrak{W}}}$

1

Preliminaries

For a topological space $X$ we use 7(X) to denote the set of all non-empty
closed subsets of $X$ endowed with the Vietoris topology . Recall thatis generated by all collections of the form$\tau_{V}$$\langle V\rangle=$

where$V${$S\epsilon F(X)$:$S\mathrm{C}\cup\sim V$

and$S\cap V\neq\emptyset$$\mathrm{W}$whenever $V\in V$ }run6 over the flnite families of open subsets of $X$. Define’$\backslash ,$$\mathcal{K}(X\rangle\overline{\sim}$.
forandIf $D\subseteq F(X)$ , then a map : )a selection on Z) provided that$f(S)\in S$ for every $S\in D$ . A selection: $Darrow X$ is
if it iscontinuous with respect to the relative Vietoris topology on 1), A selection: $F_{1}(X)arrow X$ on $F_{l}(X)$ is called a weak selection.
Suppose that is a weak selection. Then deflnes a natural order-likerelation“on $X$ , by letting
if and only if $f\langle\{x,y\})=x$ For
convenience, we will write that $x\prec y$ if
and $x\neq y$ . If and are(not necessarily non-empty) subsets of $X$ , we write $B\prec C$ provided that
for,every $y\in B$ and $z\in C$ .
$\{S\in F(X) : |S|<\omega\}$$\mathcal{F}_{n}(X)=\{S\mathrm{e}\mathcal{F}(X) : |S|\leq n\}$$f$$\mathrm{Z}$$n\epsilon\omega$$arrow X^{1}i\mathrm{s}$$continuou\mathit{8}$

$f$

$f$

$f$$”\preceq$$f$$x\prec\sim y$$y\prec z$.

$x\preceq y$

$B$

$C$

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Proposition 1.1. [4, Theorem 3.1] Let $X$ be a Hausdorff space, : $F_{2}(X)arrow$$X$ be a selection, and $ld\preceq be$ the order.like relation generated by.
thermore assume that $x,y\in X$ satisfy $x\prec y$ . Then is continuous at $\{x, y\}$if and only if there enist disjoint open sets and such that $x\in U_{f}y\in V$,.
and$f$$\mathfrak{W}r\neg$

$f$

$f$

$V$

$U$

$U\underline{\prec}V$

Given a selection on ,172(X), one can deflne a selection on $F_{2}(X)$ asfollows: $f^{*}(\{x\})=x$ for $x\in X$ and $f^{*}(\{x,y\}\rangle=y$ if and only if $f(\{x,y\})\simeq x$
whenever $x,y\in X,$ $x\neq y[4]$ . Since is a Hausdorff space, Proposition 1.1
yields that is a continuous weak selection if and only if is a continuous
the order-like relatioo generated byweak selection. We will denote.
Obviously,
if and only if$f$$f^{*}$$X$$f^{*}$$f$$f^{*}$$\mathrm{b}\mathrm{y}\underline{\prec}^{\mathrm{u}}$‘$y\sim\prec*x$$x\underline{\prec}\mathrm{y}$and a setDeflnition 1.2. [3] Given a weak selection :$F(X)$ , we will call a subset
an -maximum $(f\cdot minimum)$ of
and the following conditions hold:$f$$\mathcal{F}_{2}(X)arrow X$$f$$B\mathrm{C}\vee S$$s\epsilon$

$S$

if

$B\epsilon F(X)$(1) $S\backslash B\prec B(B\prec S\backslash B)$ ,$C\in F(X)$ , and(2) if$C\subset\sim S,$$S\backslash C\prec C(C\prec S\backslash C)$, then$B\underline{\subset}C$.

of if and only if lt is an ’-minimum
Clearly, a set is anof , so the following proposition is essentlally due to Gutev and Nogura [3].
$B$

$S$

$f_{r}\mathrm{m}\mathrm{a}\mathrm{x}\mathrm{i}\mathrm{m}\mathrm{u}\mathrm{m}$$\mathrm{f}$$S$

a weak selection
Proposition 1.3. Let be a Hausdorff space,$and\preceq the$ order-like relation generated by. Then $eve\eta$ non-empty compact
subset $S\in F(X\rangle$ has an unique -minimum and $f \cdot\max:mum$.
$X$$f:F_{\mathrm{R}}(X)arrow X$

$f$

$f$

Definition 1.4. [3] Given a weak selection :(in particular, flnite) set $s\epsilon F(X)$ , we will use $\min_{f}S$ andof , respectively.
the -minimum and$f$$f$$\mathcal{F}_{2}(X)arrow X$$\hslash\iota \mathrm{u}_{f}S$$S$$f\cdot \mathrm{m}\mathrm{a}\mathrm{x}\mathrm{i}\mathrm{m}\mathrm{u}\mathrm{m}$Proposition 1.5. Suppose that :an order-like relation generated by $f,$
and is compact. Then$f$

$F_{\mathrm{I}}(X)arrow X$

is a weak selection, 5 is
$D*E$ ,

$S,D,E\underline{\mathrm{C}}X,$$S$$S\cap D\in \mathcal{F}(X)$and a compact
to denote$\min_{f\sim}S\mathrm{C}D$.

$S\underline{\subset}D\mathrm{u}E,$

2 Extension of weak selection to finite sets
The special case of item (i) of our next lemma (whenproved in [3].

$U\overline{\sim}X$) has been

space,
its open subset, $BdU$ thea continuous weak selection, $and\preceq the$:boundary of $U,$.
order-like relation gen erated by . Let

Lemma 2.1.‘ Let
$f$

$X$

be a

$Hausdo \mathrm{r}\oint$$U$$\mathcal{F}_{l}(X)rightarrow*X$$f$$p_{u}=\{S\in \mathcal{K}(X) : S\cap U\neq\emptyset\}$29(i) If $U\prec BdU$ , then the map$U)$ is continuous.
(ii) If $BdU\prec U$ , then the map th:$U)$ is continuous.

$\varphi:\mathcal{E}_{U}arrow \mathcal{E}_{U}$$\mathcal{E}_{U}arrow \mathcal{E}_{U}$defined by$\varphi(S)=\min_{f}(S\cap$defined by $\psi(S)=\max_{f}(S\cap$

By using Lemma 2.1, we get the following main theorem.

. Let

be a continuous
be a Hausdorff space, :weak selection, and $let\preceq be$ the order-like relation generated by . $mnhe\Gamma-$.
more, assume that $p\in X$ and is a continuous selection onextending .
Then there exists a continuous selection on$\mathrm{T}\mathrm{h}\mathrm{e}\mathrm{o}\mathrm{r}\mathrm{e}\infty 2.\mathit{2}$

$X$

$f$

$\mathcal{F}_{2}(X)arrow X$$f$$\mathcal{K}(X\backslash \{p\})$

$g$

$h$

$f$

$\mathcal{K}(X)$The following is an application of Theorem 2.2.
Corollary 2.3. Let$p\in X$ , and let
to a continuous$X$

be a Hausdorff space with a single non-isolated point
be a weak sdection. Then can be extended.
on$f:F_{2}(X\rangle$ $arrow X$$\mathit{8}election$

3$f$$\mathcal{K}(X)$Selections for finite subsets of paracompact scatteredspacesLet us recall the definition of a scattered space. For every ordinal number, we define by transfinite induction the a-derivative of a space $X:X^{(0)}=X$ ;$X^{(\alpha+1)}=(X^{(\alpha)})’=$}{$x\in X$ ; is not an isolated point of$X$for some
if a is limit. A space is called scattered if$X$$X$$h(X)$ofis the smallest
ordinal or. For a scattered space , the heightis. For every $\alpha<h(X),$ ach
ordinal such thatof such that, thus there exists a neighborhood
an isolated point of$\alpha$$X^{(\alpha)}$$x$$X^{(\alpha)}=\emptyset$$\bigcap_{\beta<\alpha}X^{(\beta)}$$X^{(\alpha)}=\emptyset$$\alpha$$X^{(a)}$$V_{l}\cap X^{(\alpha)}=\{x\}$$;X^{(a\rangle}=$$e$$x\in X^{(\alpha)}\backslash X^{(\alpha+1)}$$V_{x}$.

$x$

The following lemma is part of folklore, for example [2, Lemma 2.11.

Leinma 3.1. Suppose that is a clopen partition of a space $X$ such that for.
on, there exists a continuous selection $fu$ :Then
has a continuous selection.
$\mathcal{U}$$eve\eta U\in \mathcal{U}$$\mathcal{K}(U)arrow U$$\mathcal{K}(U)$$\mathcal{K}(X)$The existence of continuous selections on scattered spaces has been
ied in $[1, 2]$ . A countable regular space has a continuous selection if and only
if it is scattered [2, Theorem 2.4]. A paracompact scattered space admits acontinuous selection provided that every point has a countable[ , Theorem 2.3], By using transfinite induction With respect to the height$h(X)$ of $X$ and applying Theorem 2.2, we can get our last theorem which
contributes to thi8 topic and provides a positive partial answer to Problem
5 from .
$\mathrm{s}\mathrm{t}\mathrm{u}\mathrm{d}\sim$$\mathrm{p}\mathrm{s}\mathrm{e}\mathrm{u}\mathrm{d}\triangleright \mathrm{b}\mathrm{a}8\mathrm{e}$$2$$\{5|$

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Theorem 3.2. If ahereditanly paracompact Hausdorff space $X$ hasa weak selection, then it also has a continuous selection on.
$\mathit{8}cattered$$\mathcal{K}(X)$Acknowledgemeot: The author thanks
fessor D. Shakhmatov for valuable comments.

$\mathrm{P}\mathrm{r}\mathrm{o}\mathrm{f}\mathrm{e}8\mathrm{s}\mathrm{o}\mathrm{r}$T. Nogura and Pro-References[1J G. Artico, U. Marconi, R. Moresco, and J. Pelant, Selectors and scatteredspace8, Topology Appl. 111 (2001), 35-48.

[2] S. FUji, K. Miyazaki, and T. Nogura, Vietoris continuous selections on
scattered spaces, J. Math. Soc. Japan 54 (2002), no. 2, 273-281.
[3| S. Garcia-Ferreira, V. Gutev, and T. Nogura, Extension of -point selections, New Zealand J. Math. (2006), accepted for publication.
$Z$[4] V. Gutev and T. Nogura, Selections and order-like relations, Applied
General Topology 2 (2001), 205-218.
[5] V. Gutev and T. Nogura, Michael’s selection problem, in: Open Problem

in Topology, II, to appear.