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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



On the dimension of $ \mu $-spaces

Author: T. Mizokami
Journal: Proc. Amer. Math. Soc. 83 (1981), 195-200
MSC: Primary 54F45
MathSciNet review: 620012
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Abstract: The main results are as follows:

Theorem 1. Let $ \mathcal{C}$ be the class of all $ \mu $-spaces. Then the following are equivalent for every $ X \in \mathcal{C}$: (1) dim $ X \leqslant n$, (2) there exists a closed mapping $ f$ of $ Z \in \mathcal{C}$ with dim $ Z \leqslant 0$ onto $ X$ such that ord $ f \leqslant n + 1$, (3) $ X = \cup _{i = 1}^{n + 1}{X_i}$, where dim $ {X_i} \leqslant n$ for each $ i$ and (4) Ind $ X \leqslant n$.

Theorem 2. A space $ X$ is a $ \mu $-space with dim $ X \leqslant n$ if and only if $ X$ is the inverse limit of an inverse sequence $ \left\{ {{X_i},g_j^i} \right\}$ of paracompact $ \sigma $-metric spaces $ {X_i}$ such that dim $ {X_i} \leqslant n$ for every $ i \in N$.

As the applications of them, some product theorems for covering dimension are given.

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Keywords: $ \sigma $-metric space, $ \mu $-space, cubic $ \mu $-space, inverse limit
Article copyright: © Copyright 1981 American Mathematical Society