A note on countable-dimensional metric spaces
A note on countable-dimensional metric spaces
复制标题
关于可数维度量空间的注记
DOI：
10.3792/pja/1195522489
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发表时间：
1965
期刊：
影响因子：
--
通讯作者：
J. H. Roberts
中科院分区：
文献类型：
--
作者：
K. Nagami;J. H. Roberts
This paper is a supplementary note to the characterization of countable-dimensional metric spaces by J. Nagata 2. A space is countable-dimensional if it is the countable sum of zero-dimensional (in the sense of the covering dimension) subsets. A space is s$rongly countable-dimensional if it is the countable sum of finite dimensional closed subsets. Now Nagata has characterized these two classes of infinite dimensional metric spaces as follows: Theorem A 2, Theorem 2.3. A metric space is countabledimensional if and only if for every collection {U: r< v} of open sets and every collection {F" <v} of closed sets such that Fc U, <v, and such hat { U" <} is locally finite for every r, there exists a collection of open sets V, <v, satisfying i) F Vo U, <r, ii) order (x, B())< o for every x e X, where --( V" and B()-{B( V)VV: < Theorem B 2, Theorem 5.3. A metric space X is strongly countable-dimensional if and only if there exists a sequence 111> 1I: >115 >1I >... of open coverings lli of X such ha i) for x e X, {St (x, lli). i-1,2,...} is a local base of x, ii) for x e X, sup order (x, 1I)< o. Our supplementary theorems to these are as follows" Theorem 1. A metric space X is countable-dimensional if and only if for every sequence of pairs of disjoint closed sets Cx, C’; C., C’;..., there exist separating closed sets B between C and C’, i-1,2,..., such that {B: i-1,2,--.} is point-finite. The only if part of this theorem is a special case of Nagata 2, Lemma 2.1. Theorem 2. A metric space X is strongly countable-dimensional if and only if there exists a sequence tl>IL.>... of open coverings 1I of X such that i) for x e X, {St(x, U): i-1,2,...} is a local base of x, ii) for x e X, sup order (x, 11)< To prove Theorem 2 we need the following theorem for finite dimensional spaces.
