Quasiorders on topological categories

Abstract

We prove that, for every cardinal number $\alpha\geq {\mathfrak c}$, there exists a metrizable space $X$ with $|X|=\alpha$ such that for every pair of quasiorders $\leq_1$, $\leq_2$ on a set $Q$ with $|Q| \leq \alpha$ satisfying the implication $$q \leq_1 q' \implies q \leq_2 q'$$ there exists a system $\{X(q) : q\in Q\}$ of non-homeomorphic clopen subsets of $X$ with the following properties: (1) $q \leq_1 q'$ if and only if $X(q)$ is homeomorphic to a clopen subset of $X(q')$, (2) $q \leq_2 q'$ implies that $X(q)$ is homeomorphic to a closed subset of $X(q')$ and (3) $\neg (q \leq_2 q')$ implies that there is no one-to-one continuous map of $X(q)$ into $X(q')$.