Compact metric spaces as minimal-limit sets in domains of bottomed sequences

Abstract

Every compact metric space $X$ is homeomorphically embedded in an $\omega$-algebraic domain $D$ as the set of minimal limit (that is, non-finite) elements. Moreover, $X$ is a retract of the set $L(D)$ of all limit elements of $D$. Such a domain $D$ can be chosen so that it has property M and finite-branching, and the height of $L(D)$ is equal to the small inductive dimension of $X$. We also show that the small inductive dimension of $L(D)$ as a topological space is equal to the height of $L(D)$ for domains with property M. These results give a characterisation of the dimension of a space $X$ as the minimal height of $L(D)$ in which $X$ is embedded as the set of minimal elements. The domain in which we embed an $n$-dimensional compact metric space $X$ ($n \leq \infinity$) has a concrete structure in that it consists of finite/infinite sequences in $\{0,1,\bot\}$ with at most $n$ copies of $\bot$.