2002/04/10 by Vera Trnkova
Mathematics · #math.GN #msc:54B30 #msc:54H10
published as Proceedings of the Ninth Prague Topological Symposium, (Prague, 2001), pp. 321--330, Topology Atlas, Toronto, 2002 · 10 pages
arxiv created 2002/04/10 · arxiv updated 2009/11/30
We prove that, for every cardinal number α≥ \mathfrak c, there exists a metrizable space X with |X|=α such that for every pair of quasiorders ≤1, ≤2 on a set Q with |Q| ≤ α satisfying the implication q ≤1 q' ⇒ q ≤2 q' there exists a system \X(q) : q∈ Q\ of non-homeomorphic clopen subsets of X with the following properties: (1) q ≤1 q' if and only if X(q) is homeomorphic to a clopen subset of X(q'), (2) q ≤2 q' implies that X(q) is homeomorphic to a closed subset of X(q') and (3) ¬ (q ≤2 q') implies that there is no one-to-one continuous map of X(q) into X(q').