数理解析研究所講究録
第 1634 巻 2009 年 55-6155

Topological Spaces of Discrete Distributions
寺田

横浜国立大学環境情報研究院

敏司 (Toshiji Terada)

Graduate School of Environment and Information Sciences,

Yokohama National University
Let be a real-valued function defined on a non-empty set . If $f$ satisfies the following
conditions, then we call that is a discrete distribution on :(1) $0\leq f(s)\leq 1$ for any $s\in S$ .
(2) $\Sigma\{f(s) : s\in S\}\leq 1$ .
In case $\Sigma\{f(s) : s\in S\}=1$ is satisfied in (2), is called to be a discrete probability
distribution on . Let the support $spt(f)$ of be the set $\{s\in S:f(s)>0\}$ . Obviously,$spt(f)$ is a countable subset offor any discrete distribution .
$S$

$f$

$S$

$f$

$f$

$S$

$f$

$S$

$f$

Let $DD(S)$ and $DPD(S)$ be the set of all discrete distributions and the set of all
discrete probability distributions on a set respectively. $DPD(S)$ is sometimes denotedby. We consider the topology of pointwise convergence on $DD(S)$ . This topology on$DD(S)$ is the relative topology induced by naturally embedding $DD(S)$ into the powerof the unit interval $I=[0,1]$ .
space, and hence $DD(S)$ is compact. $DD(S)$It is obvious that $DD(S)$ is a closed subset ofis Fr\’echet as a subspace of a -product of the unit intervals. A cardinal is consideredas the topological space with the usual interval topology.
$S$$\Sigma_{S}$$I^{|S|}$$I^{|S|}$$\Sigma$$\kappa$: $\kappaarrow[0,1]$ with $f(O)=0$ toLet us call a non-decreasing continuous function
be a cumulative, discrete distribution on . Let $CDD(\kappa)$ be the set of all cumulative,
discrete distributions on . On $CDD(\kappa)$ , we consider also the topology of pointwiseis the subspace of $CDD(\kappa)$ consisting of functions
convergence. The space CDPD
satisfing$f$$\kappa$$\kappa$$(\kappa)$$hm_{\alphaarrow\kappa}f(\alpha)=1$

Fact 1 (Dydak). A topological space $X$ is metrizable if and only if $X$ is embedded in$DPD(S)$ for some set S. In other words, for a cardind $\kappa_{l}DPD(\kappa)$ is universd for.
metrizable spaces of weight$t$$\leq\kappa$

Theorem 1 The spaces $DPD(\kappa)$ and CDPD$(\kappa)$

are homeomorphic.

56

Proof. For $f\in DPD(\kappa)$ , let $F:\kappaarrow[0,1]$ be the function defined byfor$F( O)=0,F(\alpha)=\sum\{f(\beta):\beta<\alpha\}$$0<\alpha<\kappa$.

: $DPD(\kappa)arrow CDPD(\kappa)$ defined by $\Psi(f)=F$ is obviously one-to-one
Then the map
and onto.
Claim 1.
is continuous.
$\Psi$$\Psi$$finitesubsetAofspt(f)suchthat\sum_{\circ\epsilon A}f(a)>l-\frac{\Psi\epsilon}{4}.AssumethatAiscomposedofnLetfbeanarbitrarypointinDPD(\kappa)andlet(f)=FForany\epsilon>0,th\alpha eisa$

and $g\in DPD(\kappa)$ be an arbitrary point such that $|g(a)-f(a)|<\delta$
elements. Let
for every $a\in A$ . Then$\delta=\frac{e}{4n}$$1- \frac{\epsilon}{2}<\sum_{a\in A}g(a)\leq 1$

and hence$\sum_{a\not\in A}g(a)<\frac{\epsilon}{2}$. Further, for any,

$B\subset A$$| \sum_{b\in B}g(b)-\sum_{b\in B}f(b)|\leq\sum_{b\in B}|g(b)-f(b)|\leq|B|\frac{\epsilon}{4n}<\frac{\epsilon}{4}$Then for any$\alpha<\kappa$.

,$| \sum_{\beta<\alpha}g(\beta)-\sum_{\beta<\alpha}f(\beta)|\leq|\sum_{\beta<\alpha,\beta\not\in A}g(\beta)|+|\sum_{\beta<a,\beta\not\in A}f(\beta)|+|\sum_{\beta<\alpha,\beta\epsilon A}g(\beta)-\sum_{\beta<\alpha,\beta\in A}f(\beta)|$$< \frac{\epsilon}{2}+\frac{\epsilon}{4}+\frac{\epsilon}{4}=\epsilon$.

, and henceisThis means that for $G=\Psi(g),$ $|G(\alpha)-F(\alpha)|<\epsilon$ for every
continuous.
is continuous.
Claim 2.
Let $F\in CDPD(\kappa)$ and $f=\Psi^{-1}(F)$ . Notice that $f(\alpha)=F(\alpha+1)-F(\alpha)$ . For. We can assume, there is a subset of $spt(f)$ such thatany$\alpha<\kappa$$\Psi$$\Psi^{-1}$$\epsilon>0$$A$$\sum_{a\in A}f(a)>1-\frac{\epsilon}{4}$. Suppose that $G\in CDPD(\kappa)$ satisfies thatconsists of elements. Let$|G(a)-F(a)|<\delta$ for any $a\in A\cup A+1$ , where $A+1=\{a+1 : a\in A\}$ . Then $g=\Psi^{-1}(G)$satisfies$|g(a)-f(a)|=|(G(a+1)-G(a))-(F(a+1)-F(a))|$that$A$$n$$\delta=\frac{\epsilon}{4n}$$\leq|G(a+1)-F(a+1)|+|G(a)-F(a)|<2\delta=\frac{2\epsilon}{4n}\leq\epsilon$

for any $a\in A$ . Since$\sum_{a\in A}g(a)\geq\sum_{a\in A}f(a)-n\frac{2\epsilon}{4n}>1-\frac{3\epsilon}{4}$$|g(b)-f(b)|<|g(b)|+|f(b)|< \frac{u}{4}+\frac{e}{4}=\epsilon$continuous.

,for any $b\not\in A$ . This shows that$\Psi^{-1}$is

57

We assum that all topological spaces considered here are Tychonoff. What is the classof topological spaces embedded in $DD(S)$ for some set? This is the theme of thisnote. If is an uncountable set, then the constant zero function is in $DD(S)$ and thepseudo-character at in $DD(S)$ is uncountable. That is, $DD(S)$ is not metrizable. Itis obvious that $DPD(S)$ is a dense metrizable subspace of $DD(S)$ . A space embedded
in $DD(S)$ for some set
is called a DD-space here. That is, $X$ is a DD-space if
only if there exists a familyof continuous functions $homX$ to such thatfor each $x\in X$ and the topology of $X$ coincides with the topology
induced by.
$e$

$S$

$S$

$0$

$0$

$S$

$and_{r}$$\{f_{\alpha} :$\Sigma\{f_{\alpha}(x) :\alpha\in\kappa\}$$I$\alpha\in\kappa\}\leq 1$$\{f_{\alpha} :\alpha\in\kappa\}$Theorem 2 (0) Every metrizable space is a DD-space.
(1) If is a subspce of a DD-space $X$ , thenis a DD-space.
$Y$$Y$(2) Ifis a DD-space.

$\{X_{\alpha} :$(S)$

If$\{X_{n} :\alpha\in A\}$n\in\omega\}$

is a family of DD-spaces, then the topological $sum\oplus\{X_{a} : \alpha\in A\}$is a countable family of DD-spaces, then the product space$\prod\{X_{n}$:is a DD-space.
(4) Every DD-space has a compactification which is also a DD-space.

$n\in\omega\}$

Theorem 3 Let $X$ be a DD-space. Then there is a reab valued jfUnction $\phi:Xarrow I$ suchthat the topology induced by the topology of $X$ and $\{\phi^{-1}((u,v))$ : $(u,v)$ is an open intervd
in $[0,1]\}$ is metrizable. Especially, let $\phi:DD(S)arrow I$ be the fi’ nction defined by $\phi(f)=$$1-\Sigma\{f(s) : s\in S\}$ . Then the space with the topology induced by the topologyof $DD(S)$and inverse images of open intervals by is homeomorphic to $DPD(S)$ .
$\phi$Let us recall that a compact space $K$ is called uniformly Eberlein compact if it is homeomorphic to a weakly compact subsets of a ffilbert space. The space, for a set,is defined by$c_{0}(\Gamma)$$c_{0}(\Gamma)=\{x\in R^{\Gamma}:|\{\gamma\in\Gamma:|x(\gamma)|>\epsilon\}|<\omega,\forall\epsilon>0\}$$\Gamma\neq\emptyset$.

The norm on
is the $\sup$ norm. The weak topology on a weakly compact subset ofis exactly the topology of pointwise convergence.
$c_{0}(\Gamma)$$c_{0}(\Gamma)$

Fact 2 (Benyamini-Starbird). A compact space $K$ is uniformly Eberlein compact if andonly if $K$ is homeomorphic to a subset $K$ “ offor some with the property that forevery
there exists
such that for every $x\in K’’$ ,$\Gamma$$c_{0}(\Gamma)$$\epsilon>0$$N(\epsilon)\in\omega$$|\{\gamma\in\Gamma:|x(\gamma)|>\epsilon\}<N(\epsilon)$.

58

We say that a family $A$ of subsets of a set $X$ is boundedly point finite if there exists some$n\in\omega$ such that for every $x\in X$ ord $(x, A)\leq n$ .
A. family $A$ of subsets of $X$ is said tois boundedly pointsuch that each familybe -boundedly point finite iffinite. A family $A$ of subsets of a set $X$ is called -separating if whenever $x,y\in X$ are
distinct, then some $A\in A$ contains exactly one of and .
$A= \bigcup_{k\in\omega}\mathcal{A}_{k}$$\sigma$$\mathcal{A}_{k}$$T_{0}$$x$$y$Fact 3 (Benyamini-Rudin- Wage). A compact space $K$ is uniformly Eberlein compact ifand only if $K$ has a -boundedly point finite -separating family by cozero-sets.
$T_{0}$$\sigma$

Theorem 4 The space $DD(S)$ is uniformly Eberlein compact for any set$S$.

In fact, let $Q’$ be the set of all rational numbers in $[0,1]$ . For each $q\in Q’$ and $s\in S$ ,let$U.(q)=\{f\in DD(S):f(s)>q\}$ .

Then$A=\{U.(q):s\in S\}$is a boundedly point finite famly by cozero-sets in $DD(S)$ . Further, let$A= \bigcup_{\sigma\epsilon Q’}\mathcal{A}_{q}$Then $A$ is a -boundedly point finite$\sigma$$T_{0}$.

-separating family by cozero-sets.

Theorem 5 Every uniformly Eberlein compact space is a DD-space.
Letbe a -boundedly point finite -separating family by cozero-sets in, leta uniformly Eberlein compact space $X$ . For eachbe a positive integer suchthat ord, we take a $[0,1]$ -valued continuous
for any $x\in X$ . For each$X$onwith $f_{U}^{-1}((0,1])=U$ . Further, the function
functionis defined by$A= \bigcup_{n\in\omega}\mathcal{A}_{n}$$T_{0}$$\sigma$$n\in\omega$$(x,A_{n})\leq k_{r\iota}$$k_{\iota}$$U\in \mathcal{A}$$f_{U}$$g_{U}$$g_{U}= \frac{1}{2^{n}k_{\mathfrak{n}}}f_{U}$Let $\mathcal{F}_{n}=\{g_{U}:U\in A_{n}\}$ and$\mathcal{F}=\bigcup_{n\epsilon\omega}\mathcal{F}_{n}$.

. Then the map$\Phi$: $Xarrow[0,1]^{A}$ defined by$\Phi(x)=\{g_{U}(x):U\in A\}$is a topological embedding of $X$ into$DD(\mathcal{A})$. Note that$\sum_{U\in A}g_{U}(x)=\sum_{n=1}^{\infty}\frac{1}{2^{n}k_{n}}\sum_{U\epsilon\wedge}f_{U}(x)\leq 1$.

59

Corollary 1 $DD(\kappa)$ is universd for uniformly Eberlein compact spaces of weight$\leq\kappa$.

The following two theorems may be proved under more general conditions. But, we give
direct proofs here.
Theorem 6 Let $X$ be a DD-space. If $X$ is countably compact, then $X$ is compact.

Proof. For each $r\leq 1$ , let(resp.
) be the subset of of $DD(S)$ consisting of$f$$\Sigma\{f(s):s\inS\}\leqr$
all with(resp.
: $s\in S\}<r$ ). It suffices to show that
$X$ is Lindelof. Assume that $X$ is not
Lindelof. Then there is an open cover with no$=X\cap D_{\leq}$countable subcover. Let
and $X_{<r}=X\cap D_{<r}$ for $0\leq r\leq 1$ . Then thereexists$r_{0}= \sup$ { $r:X_{\leq t}$ is covered by a countable subfamily of}.
$D_{<r}$$D_{\leq f}$$\Sigma\{f(s)$$X_{\leq}$.

.

$\mathcal{U}$$\mathcal{U}$It foUows that there exists a countable subfamily
thatis covered by , since$X_{\leq f}0$

which covers
It is also true
$X_{=\prime 0}=X_{\leq\prime}0-X_{<r0}$ is metrizable. Further, let$\mathcal{U}_{0}$$\mathcal{U}_{0}$of$X_{<ro}$$\mathcal{U}$$r_{1}= \inf\{r:(X-\cup \mathcal{U}_{0})\cap X_{\leq r}\neq\emptyset\}$Then $r_{0}=r_{1}$ must be satisfied. Let$F_{n}=X_{\leq(ro+1/n)}-\cup \mathcal{U}_{0}$is a decreasing sequence of closed subsets ofcontradicts the countable compactness of $X$ .

$X$such that.

for $n=1,2,$$\cap\{F_{n} :$\cdots$. Then thisn=1,2, \cdots\}=\emptyset$. This

Theorem 7 For a DD-space $X$ , the cardindities $c(X),d(X)$ and $w(X)$ are all the same.
Let $d(X)=\lambda$ and $D$ be a dense subset of $X$ such that $|D|=\lambda$ . Then thecardinality of $A=\cup\{spt(x):x\in D\}$ is . Since $D$ is a subset of the compact set

Proof.

$\lambda$$(DD(S)\cap I^{A})x\{0\}^{S-A}$in $DD(S),$ $X$ must be a subspace of $I^{A}x\{0\}^{S-A}$ whose weight is$\lambda$. It follows that$d(X)=w(X)$ .
Next, we will show that $d(X)\leq c(X)$ Of course, we can assume that $d(X)$ is uncountable. Let $\kappa\leq d(X)$ be an arbitrary uncountable regular cardinal. then there exists a transfinite sequenceof points in $X$ such that $spt(x_{\alpha})-\cup\{spt(x_{\beta}) : \beta<\alpha\}\neq\emptyset$ .
Further, we can fix a positive integer such that there exists $u_{\alpha}\in spt(x_{\alpha})-\cup\{spt(x_{\beta})$ :$\beta<\alpha\}$ with $x_{\alpha}(u_{\alpha})>1/k$ for any. Let $U_{\alpha}=\{x\in X : x(u_{\alpha})>1/k\}$ . Thenthe family
satisfies that each intersection of members of is empty.
Hence there must be a disjoint family consisting of non-empty open subsets.

.

$\{x_{\alpha} :\alpha<\kappa\}$$k$$\alpha<\kappa$$\mathcal{U}=\{U_{\alpha} :$k$\alpha<\kappa\}$$\mathcal{U}$$\kappa$

As mentioned previously, the spaces $DPD(\kappa),CDPD(\kappa)$ are homeomorphic for any
cardinal number . However $DD(\kappa)$ and $CDD(\kappa)$ are not homeomorphic for an uncontable cardinal . In fact, $DD(\kappa)$ is compact. On the other hand, $CDD(\kappa)$ is not compact.
$\kappa$$\kappa$

60

Moreover, let $IDD_{0}(\kappa)$ be the space of all non-decreasing $[0,1]$ -valued functions $f$ (which
need not be continuous) such that $f(O)=0$ , with the topology of pointwise convergence.
Then $IDD_{0}(\kappa)$ is a compactification of $CDD(\kappa)$ . Further,

Theorem 8 $CDD(\kappa)$ is not a DD-space for any uncountable cardind

For each$\alpha<\kappa$, let$f_{\alpha}\in CDD(\kappa)$$f_{\alpha}(\beta)=0$for$\kappa$.

be the function defined by$\beta\leq\alpha$,$f_{\alpha}(\beta)=1$

for$\beta>\alpha$.

Then
is a discrete subset of $CDD(\kappa)$ and the constant zero function
is in the closure of this set. But there is no sequence in converging to , which meansthat $CDD(\kappa)$ is not Fr\’echet. Hence $CDD(\kappa)$ is not a DD-space.
$A=\{f_{\alpha} ; \alpha\in\kappa\}$

$0$

$A$

Theorem 9 There is$a$$0$

one-to-one continuous map flvm $CDD(\kappa)$ onto $DD(\kappa)$ .

In fact, the map: $CDD(\kappa)arrow DD(\kappa)$ defined by $\Psi(F)(\alpha)=F(\alpha+1)-F(\alpha)$ is
one-to-one, onto and continuous.
$\Psi$Let us call a topological space $X$ to be a CDD-space if $X$ is homeomorphic to a subspaceof $CDD(\kappa)$ for some cardinal .
$\kappa$

Theorem 10 (1) Every metrizable space is a CDD-space.
(1) If a CDD-space $X$ is compact, then $X$ is a DD-space.
(2) If $X$ is a CDD-space, then there is a $\sigmaarrow boundedly$ point-finite,family.

$T_{0}$-separating cozero-Hence, it follows that there is a CDD-space $X$ such that every compactification of $X$ is
not a CDD-space.

参考文献[1] Y. Benyamini, M.E. Rudin, M. Wage. Continuous images of weakly compact subsetsof Banach spaces. Pacific J. Math. 70 (1977), 309-324.

[2] Y. Benyamini, T. Starbird. Embedding $weau_{y}$ compact sets into $Hd$ bert spaces. Israel
J. Math. 23 (1976), 137-141.
[3] J. Dydak. Extension Theory: the interface between set-theoretic and dgebraic topology.
Topology and Appl. 74 (1996), 225-258.

61[4] R. Engelking. Generd Topology. Helderman Verlag Berlin, 1989.
[5] F. Garcia. Expandable network and covering properties for uniform Eberlein compacta.
Topology and Appl. 153 (2006), 2886-2892.