ACTA MATHEMATICA UNIVERSITATIS COMENIANAE

Vol. LXXII, 1(2003)
p. 67 – 72

Topological Representations of Quasiordered Sets
V. Trnkova

Abstract.  We prove that for every infinite cardinal number $\alpha$ there exists a space $X$ with $|X|=\alpha$, metrizable whenever $\alpha\geq\C$, strongly paracompact whenever $\omega\leq\alpha\leq\C$, such that every quasiordered set $(Q,\leq)$ with $|Q|\leq\alpha$ can be represented by closed subspaces of $X$ in the sense that there exists a system $\{X_q|q\in Q\}$ of non-homeomorphic closed subspaces of \X\ such that $q_1\leq q_2$ if and only if $\X_{q_1}$ is homeomorphic to a subset of $\X_{q_2}.$ In fact, stronger results are proved here.

AMS subject classification:  54B30, 54H10
Keywords:  Homeomorphisms onto (closed,clopen) subspaces, quasiorders, ultrafilters on $\omega$, metrizable spaces