数理解析研究所講究録
1531 巻 2007 年 85-93

85

Numerical analysis of normal sequences of finite
open covers and Pontrjagin-Schnirelmann’s
theorem
加藤久男 (Hisao Kato), 筑波大学 (University of Tsukuba)

1

Introduction

Recently, there has been an increase in the importance of fractal sets in the sciences,and fractal dimension theory has been studied by many scientists and mathematicians(e.g. see [1], [5], [10] and [15]). Fractal dimensions depend on the metrics on spaces
and hence the analysis of metrics of the spaces are very important. In this note, we
study some properties of topological dimension, metrics and box-counting dimensionsof separable metric spaces from a point of view of general topology. In general topology,
the notion of normal sequence of open covers is one of the most useful tools for the study(e.g. see [11, 12, 13]). For example, the notion is the essence of metrizabihty of spaces(see Theorem 2.1). The key word is “normal sequence” of finite open covers. In thisnote, we study directly the numerical properties of normal sequences of”finite” open
covers on a given separable metric space $X$ and we will give another proof ofand Schnirelmann’s theorem. Furtheremore, by use of normal sequences we construct. In particular, we can construct
metrics which can control the values ofchaotic metrics with respect to the determination of the box-counting dimensions.
The methods used in this note are based on dimensional theoretical techniques in an
abstract topological space.
In fractal dimension theory, Pontrjagin and Schnirelmann [16] proved the following
fundamental result involving topological dimension $\dim X$ and (lower) box-counting
dimension din $B(X, \rho)$ for a compact metric space (X, ): For a metric on $X$ and, let$\mathrm{P}\mathrm{o}\mathrm{n}\mathrm{t}\mathrm{r}\mathrm{j}\mathrm{a}\dot{\mathrm{g}}\mathrm{n}$$\infty 1\mathrm{o}N\epsilon-1o\mathrm{g}\epsilon’$$\rho$$\rho$$\rho$$\epsilon>0$$N( \epsilon, \rho)=\min${$|\mathcal{U}||\mathcal{U}$is a finite open cover of $X$ with}$\mathrm{m}\mathrm{e}\mathrm{s}\mathrm{h}_{\rho}(\mathcal{U})\leq\epsilon$

and,$\dim_{B}(X,\rho)=\sup\{\inf\{\frac{\log N(\epsilon,\rho)}{-\log\epsilon}|0<\epsilon<\epsilon_{0}\}|0<\epsilon_{0}\}(=\lim_{\epsilonarrow}\inf_{0}\frac{\log N(\epsilon,\rho)}{-\log\epsilon})$where $|A|$ denotes the cardinality of a set{$A$$\dim X=\min$ $\dim_{B}(X,$. Then$\rho)|\rho$is a metric for $X$ }.

More generally, Bruijning ([2] or [12, p.81, Corollary]) showed that if $X$ is a separable
metric space, then$\dim X=\dot{\mathrm{m}}\mathrm{n}${$\dim_{B}(X,\rho)|\rho$is a totally bounded metric for $X$ }.

Pontrjagin and Schnirelmann proved their theorem by use of geometric arguments in an
was obtained
Euclidean space. In fact, such a metric on $X$ with$\rho$$\dim X=\mathrm{d}\dot{\mathrm{m}}_{B}(X,\rho)$

86
by use of geometric arguments (embedding arguments) on polyhedra approximationsof -dimensional sets in the $(2n+1)$ -dimensional Euclidean space $R^{2n+1}$ (see [12] and[16] .
$n$$)$2

Normal sequences of open covers

In this note, we need the following terminology and concepts. Let and be open
covers of a space $X$ . We assume that each element of any open cover of a space is not(e.g. see [11] and [13]). Suppose
an empty set. If refines , then we denote$X$
open
cover of $X$ . Then we denotethat is a subset of a space and is an$\mathcal{U}$$V$$\mathcal{V}$$\mathcal{V}\leq \mathcal{U}$$\mathcal{U}$$A$$\mathcal{U}$.

$St(A,\mathcal{U})=\cup\{U\in \mathcal{U}|U\cap A\neq\phi\}$Inductively, we define $St^{0}(A,\mathcal{U})=A,$$St^{1}(A,\mathcal{U})=St(A,\mathcal{U})$and$St^{p+1}(A,\mathcal{U})=St(St^{p}(A,\mathcal{U}),\mathcal{U})=\cup\{U\in \mathcal{U}|U\cap St^{\mathrm{p}}(A,\mathcal{U})\neq\phi\}(p\geq 1)$

We put$\mathcal{U}^{*}=\{St(U,\mathcal{U})|U\in \mathcal{U}\}$

Note that if$\mathcal{U}^{\star^{1}}=\mathcal{U}^{\star}$$\mathcal{U},$and$\mathcal{U}^{\Delta}=\{St(x,\mathcal{U})|x\in X\}$

are finite. Also, we put
andis finite, then, and. Inductively, we define$|\mathcal{U}^{*}|$$|\mathcal{U}|$$|\mathcal{U}^{\Delta}|$.

.

$\mathcal{U}^{\star^{0}}=\mathcal{U},$$\mathcal{U}^{\Delta^{0}}=$$\mathcal{U}^{\Delta^{1}}=\mathcal{U}^{\Delta}$$\mathcal{U}^{*^{\mathrm{P}+1}}=(\mathcal{U}^{\star^{\mathrm{p}}})^{\star}=\{St(W,\mathcal{U}^{\theta})|W\in u^{*^{\mathrm{p}}}\}$

and$\mathcal{U}^{\Delta^{\mathrm{p}+1}}=(\mathcal{U}^{\Delta^{\mathrm{p}}})^{\Delta}=\{St(x,\mathcal{U}^{\Delta^{\mathrm{p}}})|x\in X\}$.

. An
An open cover of $X$ is a star -refinement of an open cover of $X$ if. An open
open cover of $X$ is a delta -refinement of an open cover of $X$ if
cover of $X$ is a star-refinement of an open cover of $X$ if is a star 1-refinement of
U. An open cover of $X$ is a delta-refinement of an open cover of $X$ if is a delta.
1-refinement of U. Note that
$X$
is
Let Ui $(i=1,2, .., )$ be open covers of . Then the sequence
[13])
[12]
a
if
is
star-refinement
of
[11],
(e.g.
and
see
star-sequencea normal$u_{+1}$is called a normal delta-sequence if. Also, the sequence$u(i=1,2,\ldots,)$sequence.
is called a normalTheof
is a delta-refinement
is a normal star-sequence orsequence (e.g. see [11], [12] and [13]) if either
sequence
is called a devdopment
delta-sequence.
The
is a normalof $X$ if $\{St(x,\mathcal{U}_{i})|i=1,2, \ldots, \}$ is a neighborhood base for each point of $X$ .
The following theorem is wel known as Alexandroff-Urysohn’s metrization theorem(e.g. see [11, 12, 13]). We need some constructions of metrics in the proof of thetheorem.
$\mathcal{V}^{\Delta^{\mathrm{p}}}\leq \mathcal{U}$$\mathcal{U}$$p$$\mathcal{V}$$V^{p}\leq \mathcal{U}$$\mathcal{U}$$p$$\mathcal{V}$$\mathcal{V}$$\mathcal{U}$$\mathcal{V}$$\mathcal{U}$$\mathcal{V}$$\mathcal{V}$$\mathcal{V}\leq \mathcal{V}^{\Delta}\leq V\leq \mathcal{V}^{\Delta^{2}}$$\{u\}_{i=1}^{\infty}$$\mathrm{c}\mathrm{a}\mathrm{U}\mathrm{e}\mathrm{d}$$\mathcal{U}_{i+1}$$\mathcal{U}_{i}(i=1,2, \ldots, )$$\{\mathcal{U}_{i}\}_{i=1}^{\infty}$$\{\mathcal{U}_{i}\}_{i=1}^{\infty}$$(\star)\{u\}_{\dot{\iota}=1}^{\infty}$$\{u\}_{=1}^{\infty}.\cdot$$(\Delta)\{u\}_{i=1}^{\infty}$$x$

Theorem 2.1 (Alexandroff-Urysohn’s metrization theorem) A -space $X$ is metrizable if and only if there exists a sequenceof open covers $ofX$ such that$X$
is a normal sequence and a development of .
$T_{1}$$\{\mathcal{U}_{i}\}_{:=1}^{\infty}$$\{u\}_{=1}^{\infty}.\cdot$87
For any normal space $X$ and natural numbers and , we define the folowing
indices:(1) The function
is defined as the least natural number $m$ such that for every
open cover of $X$ with, there is an open cover of $X$ such that
and(see [14]).
$k$

$p$

$\star_{k}^{p}(X)$$V$$|\mathcal{U}|=k$$\mathcal{U}$$|\mathcal{V}|\leq m$$\mathcal{V}^{\star^{p}}\leq \mathcal{U}$(2) The function
is defined as the least natural number $m$ such that for every
open cover of $X$ with, there is an open cover of $X$ such that
and(see [14]).
$\Delta_{k}^{p}(X)$$|\mathcal{U}|=k$$\mathcal{U}$$|\mathcal{V}|\leq m$$\mathcal{V}$$\mathcal{V}^{\Delta^{p}}\leq \mathcal{U}$By$C_{m}^{k}$, we shall denote the set of all $m$-element subsets of the set 1, 2, ..,
$\{$

$k\}$

and

by

$= \frac{k!}{m!(k-m)!}$.

For natural numbers $k,m,p\geq 1$ with $k\geq m$ , we define the natural numbers.-... $\geq j_{\mathrm{p}}\geq 1\cdots$$\overline{\Delta}(k;m;p)=\Sigma_{m\geq \mathrm{j}_{1\lrcorner 2}}>$and$\star(-k;m;p)=\Sigma_{m>>\geq\ldots>\geq 1}\lrcorner.1\lrcorner.2\lrcorner.p\ldots j_{p}$.

, and in [6], Hashimoto andIn [3], Bruijning and Nagata determined the index[9,
Corollary
3.11] we determined the
Finally,.
in
Hattori determined the index, which is the key lemma of this note.
indicesandfor all $p\geq 1$ as$\Delta_{k}^{1}(X)$$\star_{k}^{1}(X)$$\Delta_{k}^{p}(X)$$\star_{k}^{\mathrm{p}}(X)$$\mathrm{f}\mathrm{o}\mathrm{U}\mathrm{o}\mathrm{w}\mathrm{s}$Lemma 2.2 Let $X$ be an infinite normal space with $\dim X=m$ and let
natu numbers. Then

$k$

and

$p$

be

$\mathrm{m}l$$\star(\sim k;k;(1/2)(3^{p}-1))=k[(1/2)(3^{\mathrm{p}}-1)+1)]^{k-1}$$\star_{k}^{p}(X)=\{$$\star(\sim k;m+1;(1/2)(3^{\mathrm{p}}-1))$,and$\tilde{\Delta}(k;k;2^{p-1})=(2^{p-1}+1)^{k}-(2^{p-1})^{k}$$\Delta_{k}^{p}(X)=\{$$\tilde{\Delta}(k;m+1;2^{p-1})$

3

,

,

,

if $k\leq m+1$
if $k\geq m+1$ .

if $k\leq m+1$
if $k\geq m+1$ .

Topological dimension and normal sequences offinite open covers

By use of Lemma 2.2, we obtain the next theorem which means that topologicaldimension is characterized in terms of the growth of the global cardinality $|u|$ ofmembers $u$ of normal sequences.

88
Theorem 3.1 Let $X$ be a separable metric space. Then(1)$\dim X=\min\{\lim\inf\frac{\log_{3}|\mathcal{U}_{i}|}{i}iarrow\infty|\{\mathcal{U}_{i}\}_{i=1}^{\infty}$

finite open covers of$X$

is a normal star-sequence ofand a development of $X$ }

and
(2)

$\dim X=\mathrm{m}\dot{\mathrm{i}}\{\lim.\inf\frac{\log_{2}|u|}{i}|arrow\infty|\{u\}_{\dot{\iota}=1}^{\infty}$

finite open covers of$X$

is a normal delta-sequence ofand a development of $X$ }.

For the another proof of Pontrjagin and Schnirelmann’s theorem, we need the followings.

Proposition 3.2 Let $X$ be a separable metric space. Then(1)$\dim X=\mathrm{m}\dot{\mathrm{i}}\{\dim_{B}(X,\rho_{*})|\rho_{\star}i\mathit{8}$an Alexandroff- $U\eta sohn’ s$ metric forX induced bywhich is a normal star-sequence of finite
open covers of $X$ and a development of $X$ },a sequence(2)$\{u\}_{i=1}^{\infty}$$\dim X=\min\{\dim_{B}(X,d_{\Delta})|d_{\Delta}$

is an Alexandroff-Urysohn’s metric forX induced by. which is a normal delta-sequence of finite
a sequence
open covers of $X$ and a development of $X$ }.
$\{u\}_{1=1}^{\infty}$

and
Let $X$ be a metrizable space and let
if the identity maps
is Lipschitz equivalent tohomeomorphisms.
Lipschitz
are)(X,$\rho_{1}$$\rho_{2}$$\rho_{2}$$\rho_{2}$be two metrics on $X$ . Thenand
: (X, )

$Id_{1}$$arrow(X,\rho_{2})$$\rho_{1}$$\rho_{1}$$Id_{2}$:$arrow(X,\rho_{1})$Proposition 3.3 Let (X, ) be a metric space such that is unded, $i.e.,$a nomal star (resp. $delta$) $- sequence$ of open covers of $X$. Suppose thatand a development of X. Then the followings are equivalent.
$\rho_{1}$$\rho_{1}$$\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}_{\rho_{1}}X<$$\{u\}_{i=1}^{\infty}\dot{i}$$\infty$(1) The Alexandroff- Urysohn’s metric

to

$(Z)$

$bo$

$\rho_{1}$.

$\rho_{2}$induced by$\{\mathcal{U}_{i}\}_{i=1}^{\infty}$

is Lipschitz equivalent

There are positive numbers $c_{2}\geq c_{1}>0$ such that for each ,$i$$\{U_{\rho_{1}}(x,c_{1}/3^{:})|x\in X\}\leq \mathcal{U}_{i}\leq\{U_{\rho_{1}}(x,c_{2}/3^{:})|x\in X\}$

(resp. $\{U_{\rho_{1}}(x,c_{1}/2^{i})|x\in X\}$ SII$\mathcal{U}_{i}\leq\{U_{\rho 1}(x,$$c_{2}/2^{1})|x\in X\}$).

The next proposition implies that for any separable metric space $X$ there is a naturalbijection from the set of all totally bounded metrics on $X$ to the set of AlexandroffUrysohn’s metrics on $X$ induced by normal sequences of finite open covers which aredevelopments of $X$ , up to Lipschitz equivalence.

89
Proposition 3.4 Let $X$ be a separable metric space and let
be a totally bounded
metric on X. Then there is a no rmal star (resp. $delta$)
of finite open
$X$
$X$
covers of such thatis a development of
andis Lipschitz equivalent
to , whereis the Alexandroff- Urysohn’s metric induced by. In particular,$\rho_{1}$$- sequence\{\mathcal{U}_{i}\}_{i=1}^{\infty}$$\{\mathcal{U}_{i}\}_{i=1}^{\infty}$$\rho_{1}$$\rho_{2}$$\{u\}_{i=1}^{\infty}$$\rho_{2}$$\dim_{B}(X, \rho_{1})=\dim_{B}(X, \rho_{2})$.

Theorem 3.5 (Pontrjagin-Schnirelmann and Bruijning’s theorem) Let $X$ be a separable metric space. Then{$\dim X=\min$ $\dim_{B}(X,$$\rho)|\rho$is a totally bounded$metr\dot{\tau}c$for }.
$X$

Proof. Put $\dim X=m$ . By Proposition 3.4, we see that if is any totally bounded
metric on $X$ , then there is a normal star-sequenceof finite open covers of $X$$X$.
$\dim_{B}(X,\rho_{1})=\dim_{B}(X,\rho_{2})$, wheresuch thatis a development of andisthe Alexandroff-Urysohn’s metric induced by. By use of this fact, we can prove
Pontrjagin-Schnirelmann and Bruijning’s theorem.
$\rho_{1}$$\{u\}_{i=1}^{\infty}$$\{u\}_{1=1}^{\infty}$$\rho_{2}$$\{u\}_{1=1}^{\infty}.$

Chaotic metrics with respect to the determination of the box-counting dimensions

4

In this section, we construct chaotic metrics with respect to the determination of
the box-counting dimensions. By Theorem 3.1, we know that for any separable metric. of finite open covers of $X$
space $X$ , there is a normal star (resp.
)(resp.
and a development of $X$ such that$\dim X)$ . We call such a normal sequencea fixndamental normal sequence of $X$ .
$\mathrm{d}\mathrm{e}\mathrm{l}\mathrm{t}\mathrm{a}$$- \mathrm{s}\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{e}\mathrm{n}\mathrm{c}\mathrm{e}\{l\mathit{4}\}_{1=1}^{\infty}$$\lim\inf_{1arrow\infty}\underline{\log}s_{i}\rfloor u_{i}\lrcorner=\dim X$$\lim\inf_{1arrow\infty:}1\mathrm{B}\mathrm{a}\rfloor \mathcal{U}\lrcorner:=$$\{u\}_{i=1}^{\infty}$

Theorem 4.1 Let $X$ be a separable metric space with $\dim X=m\geq 1$ . Suppose that.
is a fundamental normal star-sequence of. Then there a subsequence
Let be any real number with $\alpha\geq m(=\dim X)$ or
such thatof$X(i.e., \lim\inf_{1arrow\infty:}\underline{\log}\mathrm{a}\mathrm{L}u\lrcorner=\dim X.)$$\{\mathcal{U}_{i}\}_{*=1}^{\infty}$$\alpha=\infty$$a$$\{\mathcal{U}_{i_{j}}\}_{j=1}^{\infty}$$\dot{i}$$\{u\}_{i=1}^{\infty}$$[ \alpha, \infty]=\{\lim_{karrow}\inf_{\infty}\frac{\log_{3}|u_{j_{k}}|}{j_{k}}|\{j_{k}\}_{k=1}^{\infty}$

is an increasing subsequenceof natural numbers}.
Also, there is a totally bounded metric$\rho_{\alpha}$

on $X$ such that$[ \alpha, \infty]=\{\lim_{karrow}\inf_{\infty}\frac{\log N(\epsilon_{k},\rho_{\alpha})}{-\log\epsilon_{k}}|\{\epsilon_{k}\}_{k=1}^{\infty}$of positive numbers with
where

{

$N( \epsilon, \rho_{\alpha})=\min$$|\mathcal{U}||\mathcal{U}$is a decreasing sequence$karrow\infty \mathrm{h}\mathrm{m}\epsilon_{k}=0$},

is a finite open cover of $X$ Utth$mesh_{\rho_{\alpha}}(\mathcal{U})\leq\epsilon$}.

90
Remark. Let $X$ be a separable metric space with $\dim X=m\geq 1$ . Suppose thatis a fundamental normal star-sequence of $X$ . Then$\{\mathcal{U}_{i}\}_{i=1}^{\infty}$$[\dim X, \infty]=${

is a subsequence of$\lim\inf\frac{\log_{3}|\mathcal{U}_{i_{j}}|}{j}jarrow\infty|\{\mathcal{U}_{i_{j}}\}_{j=1}^{\infty}$$=\{\dim_{B}(X, \rho_{\star})|\rho_{\star}$$\{u\}_{i=1}^{\infty}$}

is the Alexandroff-Urysohn’s metric for $X$ induced

by a subsequence$\{l\mathit{4}_{j}\}_{j=1}^{\infty}$of$\{u\}_{=1}^{\infty}.\cdot\}$.

In other words, all box-counting dimensions of $X$ are generated by each fundamental
normal star-sequence of $X$ . In case of normal delta-sequence of finite open covers of$X$ , we have the following theorem.

Theorem 4.2 Let $X$ be a separable metric space with $\dim X=m\geq 1$ . Suppose thatis afimdamental normal delta-sequence
Let be any real number with $a\geq m(=\dim X)$ or $a=\infty$ . Then there is a.
such thatof$ofX(i.e_{f}.

$\{u\}_{i=1}^{\infty}$\lim\inf_{*arrow\infty}.1\circ[] 2\mathrm{L}^{\mathcal{U}}:\lrcorner i=\dim X.)$$\mathit{8}ubsequence$$a$$\{u\}_{1=1}^{\infty}$$\{\alpha_{f}\}_{j=1}^{\infty}$$[ \alpha, \infty]=\{\lim_{karrow}\inf_{\infty}\frac{\log_{2}|l\mathit{4}_{j_{\mathrm{k}}}|}{j_{k}}|\{j_{k}\}_{k=1}^{\infty}$

is an increasing subsequenceof natural numbers}.
Also, there is a totally bounded metric$d_{\alpha}$

on $X$ such that$[ \alpha, \infty]=\{\lim_{karrow}\inf_{\infty}\frac{\log N(\epsilon_{k},d_{\alpha})}{-\log\epsilon_{k}}|\{\epsilon_{k}\}_{k=1}^{\infty}$of positive numbers withis a decreasing sequence$\lim_{karrow\infty}\epsilon_{k}=0$}.

Corollary 4.3 Let $X$ be a separable metric space with $\dim X\geq 1$ . If is a real
, then there is a totally bounded metric
on $X$
number with $a\geq\dim X$ or$X$such that for any subset of with $\dim A=\dim X$ ,$a$$\alpha=\infty$$\rho_{\alpha}$$A$$[ \alpha, \infty]=\{\lim_{karrow}\inf_{\infty}\frac{\log N(\epsilon_{k},\rho_{\alpha};A)}{-\log\epsilon_{k}}|\{\epsilon_{k}\}_{k=1}^{\infty}$

is a decreasing sequence},of positive numbers witha finite open cover of with{$\lim_{karrow\infty}\epsilon_{k}=0$

where$N( \epsilon,\rho_{\alpha};A)=\min$particular,$A$$|\mathcal{U}||\mathcal{U}is$$\dim_{B}(A,\rho_{\alpha}|A)=\alpha$}. In$mesh_{\rho_{\Phi}}(\mathcal{U})\leq\epsilon$.

Corollary 4.4 Let $X$ be a separable metric space with $\dim X\geq 1$ . Suppose thatis a family of mutually disjoint closed subsets
of $X$ with $\dim A_{n}=\dim X$
for each . If is a real number with $a_{n}\geq\dim X$ or $a_{n}=\infty$ for each , then there
is a totally bounded metric on $X$ such that $dim_{B}(A_{n}, \rho|A_{n})=a_{n}$ for each .
$A$$\{A_{n}\}_{n=1}^{\infty}$

$n$

$n$

$a_{n}$$\rho$

$n$

91

5

Upper box-counting dimension
mal sequences of finite open covers$\overline{\dim}_{B}(X, \rho)$and nor-In this sectin, we study relations between upper box-counting dimension and normal
sequence of finite open covers. For a separable metric space (X, ), we consider theupper box-counting dimension of (X, ) (e.g. see [5] and [15]):$\rho$$\rho$$\overline{\dim}_{B}(X, \rho)=\lim_{\epsilonarrow}\sup_{0}\frac{\log N(\epsilon,\rho)}{-\log\epsilon}$.

Theorem 5.1 Let $X$ be a separable met space with $\dim X=m\geq 1$ . Suppose that
there is a sequencewhich is a normal star (resp. $delta$) $- sequence$ of finite open
$cover\mathit{8}ofX$ and a development $ofX$ such that
(resp.
$m\leqa\leq\beta\leq\infty$$m)$ . For any, there is a totally bounded metric
on $X$
withsuch that$r\dot{\tau}c$$\{u\}_{i=1}^{\infty}$$\lim_{iarrow\infty^{\mathrm{B}\mathrm{B}}}\iota\circ.\perp u\lrcorner|=m$$\inftyarrow\infty^{A\mathrm{a}_{i}\mathrm{L}u\lrcorner}10=$$a,\beta$$\rho_{\alpha.\beta}$$[a, \beta]=\{\lim_{karrow}\inf_{\infty}\frac{\log N(\epsilon_{k},\rho_{\alpha.\beta})}{-\log\epsilon_{k}}|\{\epsilon_{k}\}$numbers with
In particular,

is a decreasing sequence of positive$\lim_{karrow\infty}\epsilon_{k}=0$$\dim_{B}(X,\rho_{\alpha,\beta})=a\leq\beta=\overline{\dim}_{B}(X, \rho_{\alpha,\beta})$}.

.

Corollary 5.2 Let $I=[0,1]$ be the unit interval and let $X=I^{m}$ be the m-cube. which is a nornal star (resp. $delta$)$(m\geq 1)$ .
Then there is a sequence$\{l4\}_{1=1}^{\infty}$$-$

sequence of finite open covers and a development of $X$ such that
(resp.
). Moreover, for any a, urith $m\leq a\leq\beta\leq\infty$ , there is ametricon $X$ such that$\lim_{iarrow\infty}\mathrm{g}1!_{\iota}\rfloor \mathcal{U}\lrcorner_{i}=m$$\lim_{iarrow|}\infty\ovalbox{\tt\small REJECT} 10.\perp \mathcal{U}*\lrcorner=m$$\beta$$\rho_{\alpha,\beta}$$[a, \beta]=\{\lim\inf\frac{\log N(\epsilon_{k},\rho_{\alpha},\rho)}{-\log\epsilon_{k}}karrow\infty|\{\epsilon_{k}\}$numbers with
In particular,

is a decreasing sequence of positive$\lim_{karrow\infty}\epsilon_{k}=0$$\dim_{B}(X,\rho_{\alpha,\beta})=a\leq\beta=\overline{\dim}_{B}(X, \rho_{\alpha,\beta})$}.

.

For the case of $\dim X=0$ , we have the following.

Theorem 5.3 Let $X$ be an infinite -dimensional separable metric space. Then there.
which is a sequence of mutually disjoint clopen covers and a
is a sequence$X$development of such that $|u|=i$ for each $i=1,2,$ . For any a, unth $0\leq a\leq$
on $X$ such that
there is a totally bounded metric$\mathit{0}$$\{u\}_{1=1}^{\infty}$$\cdots$$\beta\leq\infty_{f}$$\rho_{\alpha,\beta}$$[a, \beta]=\{\lim\inf\frac{\log N(\epsilon_{k},\rho_{\alpha,\beta})}{-\log\epsilon_{k}}karrow\infty|\{\epsilon_{k}\}$numbers with
In particular,$\beta$is a decreasing sequence of positive$\lim_{karrow\infty}\epsilon_{k}=0$$\dim_{B}(X,\rho_{\alpha,\beta})=a\leq\beta=\overline{\dim}_{B}(X,\rho_{\alpha,\beta})$

.

}.

92
Compared with our results of this note and the Pontrjagin-Schnirelmann’ theorem,
finally we have the following problem.
Problem 5.4 Let $X$ be a separable metric space with $\dim X=m\geq 1$ . Does there exist
which is a normal star (resp. $delta$) $- sequence$ offinite open covers ofa sequence$X$ and a development(resp.
)
of $X$ such that
In
Does there exist a totally bounded metric for $X$ such that$X$$(m\geq
1)(e.g$. see [4] for theparticular, if is the Menger $m$ -dimensional compactum$m$compactum),is it true
Menger -dimensional$\{u\}_{i=1}^{\infty}$$\lim_{iarrow\infty}\underline{\log}_{2i}\rfloor \mathcal{U}\lrcorner:=m$$1\mathrm{i}_{\Phiarrow i}\infty=m\underline{\log}_{\lambda \mathrm{L}4}\mathcal{U}$$q$$\overline{\dim}_{B}(X, \rho)=\dim X\mathit{9}$$\rho$$\rho$

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Hisao Kato
Institute of Mathematics
University of Tsukuba
Ibaraki, 305-8571 Japan-mail: hisakato@sakura.cc.tsukuba.ac.jp