Quasiorders on topological categories

Vera Trnková

We prove that, for every cardinal number a greater than or equal c, there exists a metrizable space X with |X|=a such that for every pair of quasiorders <= ₁, <= ₂ on a set Q with |Q| = a satisfying the implication

q <= 1 q' implies q <= 2 q'

there exists a system { X(q) : q in Q} of non-homeomorphic clopen subsets of X with the following properties:

• q <= ₁ q' if and only if X(q) is homeomorphic to a clopen subset of X(q'),
• q <= ₂ q' implies that X(q) is homeomorphic to a closed subset of X(q') and
• not (q <= ₂ q') implies that there is no one-to-one continuous map of X(q) into X(q').

Mathematics Subject Classification. 54B30 54H10.
Keywords. homeomorphism onto clopen subspace, onto closed subspace, quasiorder, metrizable spaces.

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arXiv

math.GN/0204143

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