Topology Atlas
Document # ppae29
Quasiorders on topological categories
Vera Trnková
Proceedings of the Ninth Prague Topological Symposium
(2001)
pp. 321330
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 nonhomeomorphic 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 onetoone
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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Copyright © 2002
Charles University and
Topology Atlas.
Published April 2002.