Quasiorders on topological categories (Q2782230)

The author proves that for every cardinal number \(\alpha \geq {\mathfrak c}\), there exists a metrizable space \(X\) with \(|X |= \alpha\) such that for every pair \(\leq_1\), \(\leq_2\) of quasiorders on a set \(Q\) with \(|Q|\leq \alpha\) satisfying the implication, \(q\leq_1q'\) 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:NEWLINENEWLINENEWLINE(1) \(q\leq_1q'\) if and only if \(X(q)\) is homeomorphic to a clopen subset of \(X(q'),\)NEWLINENEWLINENEWLINE(2) \(q\leq_2 q'\) implies that \(X(q)\) is homeomorphic to a closed subset of \(X(q')\), andNEWLINENEWLINENEWLINE(3) \( \neg (q\leq_2 q')\) implies that there is no one-to-one continuous map of \(X(q)\) into \(X(q').\)NEWLINENEWLINENEWLINEAmong other things it follows from this result that for every cardinal number \(\alpha \geq {\mathfrak c}\) there exists a metrizable space \(X\) with \(|X |=\alpha\) such that every quasi-ordered set \((Q,\leq)\) with \(|Q|\leq \alpha\) has an \({\mathcal M}\)-representation by clopen subspaces of \(X\) whenever \({\mathcal M}\) is the class of all continuous bijections. The author also discusses various related results from the literature, mentions some unsolved problems in the area and announces a further theorem whose proof will be published elsewhere.NEWLINENEWLINEFor the entire collection see [Zbl 0983.00046].