Relationships between usual and approximate inverse systems

Abstract

We shall prove that if $\bold X = \{X_{a}, p_{ab}, A\}$ is an approximate inverse system of compact non-metric spaces with surjective bonding mappings $p_{ab}$ such that each $X_{a}$ is a limit of a usual $\tau $-directed inverse system $X(a)=\{X_{(a,\gamma)}$, $f_{(a,\gamma)(a,\delta)}$, $\Gamma_{a}\}$ of metric compact spaces, then there exist: 1) a usual $\tau$-directed inverse system $X_{D} = \{X_{d}, F_{de}, D\}$ whose inverse limit $X_{D}$ is homeomorphic to $X=\lim\bold X$, 2) every $X_{d}$ is a limit of an approximate inverse system $\{X_{(a,\gamma_{a})}$, $g_{(a,\gamma_{a})(b,\gamma_{b})},A\}$ of compact metric spaces $X_{(a,\gamma_{a})}$,3) if the mappings $p_{ab}$ and $f_{(a,\gamma)(a,\delta)}$ are monotone, then $g_{(a,\gamma_{a})(b,\gamma_{b})}$ and $F_{de}$ are monotone.