On the dimension of $\mu$-spaces

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.