On the countable sum of zero-dimensional metric spaces

Universal spaces for countable-dimensional metric spaces

“…Universal spaces for countable dimensional metric spaces were studied by J. Nagata and it was shown that (a) the subspace Aw of the Hubert cube ¿w consisting of all points which have at most finitely many rational coordinates is a universal space for countable dimensional separable metric spaces [3,Corollary 4.4], and (b) the subspace Koo(A) of the Cartesian product P(A) of No copies of the star space S (A) is a universal space for countable dimensional metric space with weight < \A\, where \A\ denotes the cardinality of A and Koo(A) consists of all points in P(A) which have only finitely many rational coordinates distinct from zero [4,…”

Invariance of infinite-dimensional classes of spaces

“…A. V. Arhangel'skii [1] and J. Nagata [6] have developed analogues to Theorems 10 and 11 for the class of countable-dimensional spaces.…”

On the strongly countable-dimensionality of μ-spaces

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Nagata in [3] defined strongly countable-dimensional spaces which are the countable union of closed finite-dimensional subspaces. Walker and Wenner in [7] characterized such metric spaces as follows: a space X is a strongly countable-dimensional metric space if and only if there exists a finite-to-one closed mapping of a zero-dimensional metric space onto X with weak local order.

In this paper, we consider strongly countable-dimensionality for the class of n-spaces in the sense of Nagami [5] and show that the above characterization is generalized to this class.

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