数理解析研究所講究録
第 1884 巻 2014 年 48-54

48

On a function space with the hypograph topology
Katsuhisa Koshino

Graduate School of Pure and Applied Sciences,
University of Tsukuba

1

Introduction

The study of topologies on function spaces plays a significant role in geometric functional analysis.
Since function spaces are frequently infinite-dimensional, the theory of infinite-dimensional topology
has made meaningful contributions to it. Indeed, several function spaces have been shown to behomeomorphic to typical infinite-dimensional spaces. From the end of $1980s$ to the beginning of$1990s$ , many researchers investigated topological types of function spaces of real-valued continuousfunctions on countable spaces endowed with the pointwise convergence topology, see [8]. In thisarticle, we define a hypograph of a map from a compact metrizable space to a dendrite and discuss
the topology of the hypograph space. We can consider that hypograph spaces give certain geometricaspect to function spaces with the pointwise convergence topology. This article is a r\’esum\’e of thejoint work with K. Sakai and H. Yang [6].
Throughout the article, all maps are continuous, but functions are not necessarily continuous.
Let $X$ be a compact metrizable space and be a dendrite with an end point . Recall that a dendrite
is a Peano continuum, namely a connected, locally connected, compact metrizable space, containing
no simple closed curves. An end point of a space has an arbitrarily small open neighborhood whoseboundary is a singleton. It is well-known that each pair of distinct points of a dendrite is connectedby the unique arc [12, Chapter V, (1.2)]. We denote the unique arc of two points $x,$ in the dendriteby $[x, y]$ , where it is the constant path if $x=y.$of as follows:
For each function : $Xarrow Y$ , we define the hypograph
$Y$

$0$

$y$

$Y$

$f$

$\downarrow f$$f$$\downarrow f=\bigcup_{x\in X}\{x\}\cross[0, f(x)]\subset X\cross Y.$

is closed in $X\cross Y$ . We denote the set of maps from $X$
When is continuous, the hypographto by $C(X, Y)$ and the hyperspace of non-empty closed sets in $X\cross Y$ endowed with the Vietoris
topology by Cld $(X \cross Y)$ . Then we have$f$$\downarrow f$$Y$$\downarrow C(X, Y)=\{\downarrow f|f\in C(X, Y)\}\subset$

Cld $(X\cross Y)$ .

in Cld $(X \cross Y)$ . In the case that $Y$ is the closed unitLet
be the closure ofinterval $I=[O, 1]$ and $0=0$ , we can regard$\downarrow C(X, Y)$$\overline{\downarrow C(X,Y)}$$\downarrow USC(X, I)=$as the subspace in Cld
$0\}$

. Let

{

$\downarrow f|f$

: $Xarrow I$ is upper semi-continuous}

be the Hilbert cube and. Z. Yang and X. Zhou [10, 11] showed the following theorem:$(X\cross I)$$Q=I^{\mathbb{N}}$$c_{0}=\{(x_{i})_{i\in \mathbb{N}}\in Q|\lim_{iarrow\infty}x_{i}=$49
THEOREM 1.1. Suppose that the set of isolated points of $X$ is not dense.
and the pairis homeomorphic to $(Q, c_{0})$ .

For spaces, the symbol $(W_{1}, W_{2})$ means that
andis homeomorphic to
if there exists a homeomorphism
We generalize their result as follows:$W_{1}$$W_{2}\subset W_{1}.$$W_{2}$$(Z_{1}, Z_{2})$$f$:$USC$ $(X, I)=$

pair $(W_{1}, W_{2})$ of spaces$A$$W_{1}arrow Z_{1}$MAIN THEOREM. If $X$ is infinite and locally connected, then the pairhomeomorphic to $(Q, c_{0})$ .

2$Then\downarrow$$(\downarrow USC(X, I), \downarrow C(X, I))$$\downarrow C(X, I)$such that $f(W_{2})=Z_{2}.$$(\overline{\downarrow C(X,Y)}, \downarrow C(X, Y))$

isPreliminaries

The topological characterizations for pairs of infinite-dimensional spaces goes back to the uniquenessof cap sets and f-d cap sets due to R.D. Anderson [1], and now, has reached the one of absorbing
pairs for each Borel class, refer to [2, 3]. In this section, we shall introduce the notion of strong
universality and absorbing pair for the proof of the main theorem. For each open cover of a space, a map : $Warrow Z$ is -close to $g:Warrow Z$ provided that for any $w\in W$ , both of $f(w)$ and $g(w)$are contained in some
, a map : $Warrow Z$. When $Z=(Z, d)$ is a metric space, for each$WarrowZ$is said to be -close to :if $d(f(w), g(w))<\epsilon$ for all $w\in W$ . Let
be a pair ofspaces, andandbe classes of spaces. We say thatis strongly-universal if
the following condition holds:$\mathcal{U}$

$Z$

$f$

$\mathcal{U}$$\epsilon>0$$U\in \mathcal{U}$$C_{1}$$f$$(W_{1}, W_{2})$$g$$\epsilon$$(W_{1}, W_{2})$$C_{2}$$(C_{1}, C_{2})$(su) Let $Z_{1}\in C_{1},$ $Z_{2}\in C_{2},$ $K$ a closed subset of, and: $Z_{1}arrow W_{1}$ a map such that the$Z$$K$
restrictionof, there exists a
is a -embedding. Then for every open coverof$Z$-embedding $g:Z_{1}arrow W_{1}$ such that is -close to $f,$ $g|_{K}=f|_{K}$ and $g^{-1}(W_{2})\backslash K=Z_{2}\backslash K.$$f$$Z_{1}$$f|_{K}$$W_{1}$$\mathcal{U}$$g$$\mathcal{U}$It is said that a closed subset of $W$ is a $Z$ -set in $W$ if for each open cover of $W$ , there exists a
map : $Warrow W$ such that is -close to the identity map id and $f(W)\cap A=\emptyset.$countable$Z$
union of -sets is called a-set. In addition, a -embedding is an embedding whose image is a$Z$pair $(W_{1}, W_{2})$ is-set.
-absorbing provided that the following conditions are satisfied:
$A$

$f$

$f$

$\mathcal{U}$$\mathcal{U}$

$A$

$A$

$W$

$Z$

$Z_{\sigma}$$(C_{1}, C_{2})$(1)$W_{1}\in C_{1}$(2)$W_{2}$

and$W_{2}\in C_{2}$is contained in a(3) $(W_{1}, W_{2})$ is strongly;$Z_{\sigma}$-set in$(C_{1}, C_{2})$$W_{1}$;-universal.

Denote the class of compact metrizable spaces by, and the one of separable metrizable absolute-spaces by. According to Theorem 1.7.6 of [3], the following can be established.
$\mathcal{M}_{0}$$F_{\sigma\delta}$$\mathcal{F}_{\sigma\delta}$

THEOREM 2.1. Let
are$(Z_{1}, Z_{2})$andbe topological copies of the Hilbert cube $Q$ .
-absorbing, then they are homeomorphic.

$W_{1}$$(\mathcal{M}_{0}, \mathcal{F}_{\sigma\delta})$$Z_{1}$

If pairs

The following fact is well known.
FACT 1. The pair $(Q, c_{0})$ is-absorbing.
Combining Theorem 2.1 with Fact 1, we need to show the following conditions:$(\mathcal{M}_{0}, \mathcal{F}_{\sigma\delta})$(1)$\overline{\downarrow C(X,Y)}$

is homeomorphic to $Q$ and(2)$\downarrow C(X, Y)$

is contained in a(3)$(\overline{\downarrow C(X,Y)}, \downarrow C(X, Y))$

$Z_{\sigma}$-set inis strongly$\downarrow C(X, Y)$$\overline{\downarrow C(X,Y)}$$(\mathcal{M}_{0}, \mathcal{F}_{\sigma\delta})$

is an$F_{\sigma\delta}$;-universal.

-set in$\overline{\downarrow C(X,Y)}$;$(W_{1}, W_{2})$

and

50
The space3

is homeomorphic to the Hilbert cube$\downarrow C(X, Y)$

This section is devoted to proving the following theorem:

THEOREM 3.1. If $X$ has no isolated points, then$\overline{\downarrow C(X,Y)}$

is homeomorphic to $Q.$

First, we have the following proposition:

PROPOSITION 3.2. If $X$ has no isolated points, then$\overline{\downarrow C(X,Y)}$

is an $AR.$

is a Peano continuum. According to the the Wojdyslawski
Sketch of proof. Observe that
Theorem [13], see Theorem 5.3.14 of [7], the hyperspace Cld
is an $AR$ . Then we have theretraction: Cld Cld
Cld $(X\cross Y)$$\overline{\downarrow C(X,Y)}$$(\downarrow C(X, Y))$$\cup$

and(Cld
implies that$(\overline{\downarrow C(X,Y)})$$\cup$$\downarrow C(X, Y)$)$($$=\overline{\downarrow C(X,Y)}$

is an $AR.$$(X\cross Y))\ni \mathcal{A}\mapsto\cup \mathcal{A}\in$. It follows that$\overline{\downarrow C(X,Y)}$

is a retract of Cld$(\overline{\downarrow C(X,Y)})$, which$\square$We say that a subset is homotopy dense in a space $W$ if there exists a homotopy $h:W\cross Iarrow W$
such that $h(w, 0)=w$ and $h(w, t)\in Z$ for every $w\in W$ and $t>0$ . Using the same technique as [5,Theorem 4.1], we have the following:$Z$

PROPOSITION 3.3. If $X$ has no isolated points, $then\downarrow C(X, Y)$ is homotopy dense inLet
be admissible metrics on $X$ andand
on $X\cross Y$ as follows:$d_{X}$$d_{Y}$$Y$$\overline{\downarrow C(X,Y)}.$, respectively. We use an admissible metric$\rho((x, y), (x’, y’))=\max\{d_{X}(x, x’), d_{Y}(y, y’)\}$

for each$x,$ $x’\in

X$

and

$y,$ $y’\in$\rho$Y.$

inducedSince $X$ and are compact, the hyperspace Cld $(X\cross Y)$ admits the Hausdorff metricby . For each $A\in$ Cld $(X\cross Y)$ , we define a set-valued function $A:Xarrow$ Cld
ae follows:$Y$$\rho_{H}$$(Y)U\{\emptyset\}$$\rho$$A(x)=\{y\in Y|(x, y)\in A\}\in$ Cld $(Y)\cup\{\emptyset\}.$

The following is the key lemma of this article.

LEMMA 3.4 (The Digging Lemma). Let : $Zarrow\downarrow C(X, Y)$ be a map of a paracompact Hausdorff, then for each map : $Zarrow(O, 1)$ , there exist maps
space Z. If $X$ has a non-isolated point:
and : $Zarrow(O, 1)$ such that for each $z\in Z,$$\phi$$x_{\infty}$$\psi$$Zarrow\downarrow C(X, Y)$$\epsilon$$\delta$(a) $\rho_{H}(\phi(z), \psi(z))<\epsilon(z)$ ,(b) $\psi(z)(x)=\{0\}$ for all $x\in X$ with $d_{X}(x, x_{\infty})<\delta(z)$ .

A space has the disjoint cells property provided that for any maps $f,$ $g:Qarrow Z$ of the Hilbert
cube and any open cover of , there exist maps $f’,$ : $Qarrow Z$ such that $f’$ and are -close toand , respectively, and $f’(Q)\cap g’(Q)=\emptyset.$$Z$$\mathcal{U}$

$f$

$Z$

$g’$

$g’$

$\mathcal{U}$$g$

PROPOSITION 3.5. If $X$ has no isolated points, then$\overline{\downarrow C(X,Y)}$

has the disjoint cells property.

51
is homotopy dense. Sincebe maps andSketch of proof. Let $f,$ :by Proposition 3.3, we can obtain maps $f’$ : $Qarrow\downarrow C(X, Y)$ that is -close to , andand applying the: $Qarrow\downarrow C(X, Y)$ that is-close to . Taking a non-isolated point$g”$
and-close toisDigging Lemma 3.4, we can find a map $g”:Qarrow\downarrow C(X, Y)$ such that$g”(z)(x_{\infty})=\{0\}$ for all $z\in$ Q. Define a map $9”’:Qarrow\downarrow C(X, Y)\backslash \downarrow C(X, Y)$ as follows:$\epsilon>0$$Qarrow\overline{\downarrow C(X,Y)}$

$g$

in

$\downarrow C(X, Y)$$\downarrow C(X, Y)$$f$$\epsilon$$g’$$\epsilon/3$$x_{\infty}\in X$

$g$

$g’$

$\epsilon/3$$g”’(z)=g”(z)\cup\{x_{0}\}\cross\{y\in Y|d_{Y}(y, 0)\leq\epsilon/3\}.$

Then $f’$ and $g”’$ are -close to$\epsilon$

and , respectively, and $f’(Q)\cap g"’(Q)=\emptyset.$$f$$\square$$g$

Combining Propositions 3.2 and 3.5 with Toru\’{n}czyk’s characterization of the Hilbert cube [9],we can obtain Theorem 3.1.

The space

4

is an

$\downarrow C(X, Y)$$F_{\sigma\delta}$-set in$\downarrow C(X, Y)$

In this section, we show the following proposition:
PROPOSITION 4.1. The space.

Sketch of proof. For each$\epsilon>0$, define$F_{\sigma\delta}$-set in$\overline{\downarrow C(X,Y)}.$$\mathcal{A}(\delta, \epsilon)\subset\overline{\downarrow C(X,Y)}$

as follows:$x_{2}\in X$ with $d_{X}(x_{1}, x_{2})<\delta$ , if $y_{i}\in A(x_{i})$ andprovided that for each
$i=1,2$ , then$A\in \mathcal{A}(\delta, \epsilon)$

for any$\delta,$is an$\downarrow C(X, Y)$$x_{1},$

Then it is closed in$\overline{\downarrow C(X,Y)}$and we have$\downarrow C(X, Y)=\bigcap_{n\in \mathbb{N}}\bigcup_{m\in \mathbb{N}}\mathcal{A}(1/m, 1/n)$

Hence

5

$\downarrow C(X, Y)$

is an

The space$F_{\sigma\delta}$$y_{i}\not\in[0, z_{i}]$$d_{Y}(y_{1}, y_{2})\leq\epsilon.$$z_{i}\in A(x_{i})\backslash \{y_{i}\},$-set in$\downarrow C(X, Y)$$\overline{\downarrow C(X,Y)}.$$\square$is contained in a

We use the following lemma for detecting.

$Z$-sets in$Z_{\sigma}$-set in$\downarrow C(X, Y)$$\overline{\downarrow C(X,Y)}.$LEMMA 5.1. Suppose that $F=E\cup Z$ is a closed set insuch that $Z$ is a -set inand for each $A\in E$ , there exists a point $a\in X$ with $A(a)=\{0\}$ . Then $F$ is a -set $in\downarrow C(X, Y)$ .
$Z$$\overline{\downarrow C(X,Y)}$$\overline{\downarrow C(X,Y)},$$Z$PROPOSITION 5.2. If $X$ has no isolated points, $then\downarrow C(X, Y)$ is contained in some$\downarrow C(X, Y)$$Z_{\sigma}$-set in.

Sketch of proof. Take a countable dense set $D=\{d_{n}|n\in \mathbb{N}\}$ in $X$ . For each $n,$$m\in \mathbb{N},$$F_{n,m}=\{\downarrow f\in\downarrow C(X, Y)|d_{Y}(f(d_{n}), 0)\geq 1/m\}$is a $Z$ -set inbecauseis a -set indue to the Digging Lemma 3.4. Then the closure
by Proposition 3.3. $Mo$ reover, we haveis homotopy dense in$\downarrow C(X, Y)$$\downarrow C(X, Y)$$\overline{F_{n,m}}$$\downarrow C(X, Y)$$F= \bigcap_{n\in \mathbb{N}}\bigcap_{m\in \mathbb{N}}(\downarrow C(X, Y)\backslash F_{n,m})=\{X\cross\{0\}\}.$

It follows from Lemma 5.1 that the closure$\overline{F}$is a $Z$-set in$\overline{\downarrow C(X,Y)}.$$\square$$Z$$\overline{\downarrow C(X,Y)}$52
6

The pair$(\downarrow C(X, Y), \downarrow C(X, Y))$is strongly

We needs the following lemma to verify the strong$(\mathcal{M}_{0}, \mathcal{F}_{\sigma\delta})$$(\mathcal{M}_{0}, \mathcal{F}_{\sigma\delta})$-universality of-universal$(\overline{\downarrow C(X,Y)}, \downarrow C(X, Y))$.

LEMMA 6.1. Let, such thatis a strictly decreasing
sequence conversing to , and letsuch that $d_{Y}(0, y_{0})\leq 1$ . Suppose that $g:Zarrow Q$ is
an injection from a space into the Hilbert cube and : $Zarrow(O, 1)$ is a map. Then there exists
a map :
satisfying the following conditions:$x_{m},$ $x_{\infty}\in X,$$0$$m\in \mathbb{N}$$\{r_{m}=d_{X}(x_{m}, x_{\infty})\}_{m\in \mathbb{N}}$$y_{0}\in Y\backslash \{O\}$$Z$$\Phi$(1)$\Phi$$Q$$\delta$$Zarrow\downarrow C(X, [0, y_{0}])$is injective;(2) $\rho_{H}(\Phi(z), X\cross\{O\})\leq\delta(z)$ for all $z\in Z$ ;(3) $\Phi(z)(x)=\{0\}$ for all $x\in X$ with $d_{X}(x, x_{\infty})\geq r_{2k}$ and $z\in Z$ with $2^{-k}\leq\delta(z)\leq 2^{-k+1},$;$k\in \mathbb{N}$(4) $z\in g^{-1}(c_{0})$ if and only if $\Phi(z)\in\downarrow C(X, [0, y_{0}])$ ;(5) $\Phi(z)(x_{\infty})=\{y\in[0, y_{0}]|d_{Y}(y, 0)\leq\delta(z)\}$ for all $z\in Z.$

PROPOSITION 6.2. If $X$ has no isolated points, then$(\overline{\downarrow C(X,Y)}, \downarrow C(X, Y))$is strongly$(\mathcal{M}_{0}, \mathcal{F}_{\sigma\delta})-$universal.

$K$ a closed subset of $Z,$
Sketch of proof. Let
and :
a map
$Zarrow\downarrow
C(X,Y)$
such thatis a -embedding. We shall construct a -embedding :so thatis -close toand, we may assume that
Since $\Phi(K)$ is a -set in. Define $\delta(z)=$$\min\{\epsilon, \rho_{H}(\Phi(z), \Phi(K))\}/4$ .
is homotopy dense in
by Proposition 3.3,
Since
there exists $h:Zarrow\downarrow C(X, Y)$ such that $\rho_{H}(h(z), \Phi(z))\leq\delta(z)$ and.
Take a non-isolated point $x_{\infty}\in X$ . By the Digging Lemma 3.4, we can obtain:$Karrow\downarrow C(X, Y)$ and $r:Z\backslash Karrow(O, 1)$ so that
$Z\in \mathcal{M}_{0},$$\Phi,$$\epsilon$$0<\epsilon$$C\in \mathcal{F}_{\sigma\delta},$$Z$$\Phi|_{K}$$\Phi$$Z$$\Psi|_{K}=\Phi|_{K}$$Z$$Zarrow\overline{\downarrow C(X,Y)}$$\Psi$$\Psi$$\Psi^{-1}(\downarrow C(X, Y))\backslash K=C\backslash K.$$\Phi(K)\cap\Phi(Z\backslash K)=\emptyset$$\downarrow C(X, Y)$$\downarrow C(X, Y)$$\downarrow C(X, Y)$$h(Z\backslash K)\subset\downarrow C(X, Y)$$\psi$$Z\backslash$(a) $\rho_{H}(h(z), \psi(z))\leq\delta(z)$ ,(b) $\psi(z)(x)=\{0\}$ for all $x\in X$ with $d_{X}(x, x_{\infty})<r(z)$ .

Let $Z_{k}=\{z\in Z|2^{-k}\leq\delta(z)\leq 2^{-k+1}\}\subset Z\backslash K$. Since
is a non-isolated point, we$r_{m}=d_{X}(x_{m},x_{\infty})<\min\{1/m,d_{X}(x_{7n-1},
x_{\infty}), r(z)|z\in Z_{m}\}.$
can chooseso that$(Q,c_{0})$
Since
is strongly-universal by Fact 1, we can take am embedding : $Zarrow Q$$g^{-1}(c_{0})=C$
so that. Choosewith $d_{Y}(0, y_{0})\leq 1$ . Using Lemma 6.1, we can obtain:satisfying the following conditions:$x_{\infty}$$x_{m}\in X\backslash \{x_{\infty}\}$$(\mathcal{M}_{0}, \mathcal{F}_{\sigma\delta})$$g$$y_{0}\in Y\backslash \{O\}$$\psi’$$Z\backslash Karrow\downarrow C(X, [0, y_{0}])$(1)$\psi’$

is injective;(2) $\rho_{H}(\psi’(z), X\cross\{O\})\leq\delta(z)$ for all $z\in Z\backslash K$ ;(3) $\psi’(z)(x)=\{0\}$ for all $x\in X$ with $d_{X}(x, x_{\infty})\geq r_{2k}$ and $z\in Z_{k},$(4) $z\in C\backslash K$ if and only if $\psi’(z)\in\downarrow C(X, [0, y_{0}])$ ;(5) $\psi’(z)(x_{\infty})=\{y\in[O, y_{0}]|d_{Y}(y, 0)\leq\delta(z)\}$ for all $z\in Z\backslash K.$

Define$\Psi|_{Z\backslash K}$

by $\Psi(z)=\psi(z)\cup\psi’(z)$ .

$\square$$k\in \mathbb{N}$

;

53
7

Remarks

In this section, we will give some remarks on the main theorem. For more details, refer to [4].
Z. Yang and X. Zhou [11] proved the stronger result as follows:

THEOREM 7.1. The pair$(\downarrow USC(X, I), \downarrow C(X, I))$of isolated points of $X$ is not dense.

is homeomorphic to $(Q, c_{0})$ if and only if the set

It is unknown whether the same result holds or not in the general case. However, the author [4]shows the following theorem (Z. Yang [10] proved the case that $Y=I$ ).

THEOREM 7.2. The spacedense.

$\downarrow C(X, Y)$

is a Bare space if and only if the set of isolated points of $X$ is

Sketch of proof. The “only if” part follows from the same argument as Section 5. In fact, if the setof isolated points of $X$ is not dense, thenis a-set in itself, and hence it is not a Bare
space.
Next, we show the “if” part. Let
be the set of isolated points in $X$ andbe the finite
subsets of . For each
, we define
and$\downarrow C(X, Y)$$Z_{\sigma}$$X_{0}$$X_{0}$$F\in \mathcal{F}$$n\in \mathbb{N}$$U_{F,n}=\{A\in\overline{\downarrow C(X,Y)}|d_{Y}(y, 0)<1/n$Then$U_{F,n}$is open in

Baire space.

for all$x\in X\backslash F$

andis dense in
is a Baire space and dense in$\overline{\downarrow C(X,Y)}$$G= \bigcap_{n\in \mathbb{N}}U_{n}\subset\downarrow C(X, Y)$$\mathcal{F}$$U_{n}= \bigcup_{F\in \mathcal{F}}U_{F,n}$

and $y\in A(x)\}.$$\overline{\downarrow C(X,Y)}$$\downarrow C(X, Y)$. Observe that the. Consequently,$G_{\delta}$$\downarrow C(X, Y)$

-set
is a

$\square$

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Doctoral Program in Mathematics, Graduate School of Pure and Applied Sciences,
University of Tsukuba,
Tsukuba, 305-8571, JapanE–mail: kakoshino@math.tsukuba.ac.jp