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
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