Closed images of countable-dimensional spaces
Keiô Nagami · Journal of the Mathematical Society of Japan · 1967
A metric space $X$ is called countable-dimensional or $\sigma_{0}$ if $X$ is the sum of subsets $X_{i},$ $i=1,2,$ $\cdots$ , with $\dim X_{i}\leqq 0$ , where $\dim X_{i}$ denotes the covering dimension of $X_{i}$ defined by means of finite open coverings.A metric space is called uncountable-dimensional if it is not $\sigma_{0}$ .The purpose of this paper is to prove the following:, is dense-in-itself and non-empty.This was proved firstly by E. Sklyarenko [3] for the case when $X$ is compact metric and generalized by A. Arhangelskii [1] to some class of spaces which contains all separable metric spaces.But Arhangelskii's generalization is not effective for general metric spaces yet.We need three lemmas.This was proved by K. Morita-S.Hanai [2] and by A. H. Stone [4].LEMMA 2. Let $X$ be a metric space which is locally $\sigma_{0}$ .Then $X$ is $\sigma_{0}$ .PROOF.Let $\mathfrak{U}$ be a $\sigma$ -discrete base of $X$ and $\mathfrak{U}^{\prime}$ be the set of all elements $U$ of $\mathfrak{U}$ such that $U$ is $\sigma_{0}$ .Since $\mathfrak{U}^{\prime}$ is a a-discrete open covering of $X$ , we