TY - JOUR
T1 - On HCO spaces. An uncountable compact T2 space, different from ℵ1+1, which is homeomorphic to each of its uncountable closed subspaces, which is homeomorphic to each of its uncountable closed subspaces
AU - Bonnet, Robert
AU - Shelah, Saharon
PY - 1993/10
Y1 - 1993/10
N2 - Let X be an Hausdorff space. We say that X is a CO space, if X is compact and every closed subspace of X is homeomorphic to a clopen subspace of X, and X is a hereditarily CO space (HCO space), if every closed subspace is a CO space. It is well-known that every well-ordered chain with a last element, endowed with the interval topology, is an HCO space, and every HCO space is scattered. In this paper, we show the following theorems: Theorem (R. Bonnet): Every HCO space which is a continuous image of a compact totally disconnected interval space is homeomorphic to β+1 for some ordinal β. Every HCO space of countable Cantor-Bendixson rank is homeomorphic to α+1 for some countable ordinal α. Theorem (S. Shelah): Assume(Formula presented.). Then there is a HCO compact space X of Cantor-Bendixson rank ω1} and of cardinality ℵ1 such that: X has only countably many isolated points, Every closed subset of X is countable or co-countable, Every countable closed subspace of X is homeomorphic to a clopen subspace, and every uncountable closed subspace of X is homeomorphic to X, and X is retractive. In particular X is a thin-tall compact space of countable spread, and is not a continuous image of a compact totally disconnected interval space. The question whether it is consistent with ZFC, that every HCO space is homeomorphic to an ordinal, is open.
AB - Let X be an Hausdorff space. We say that X is a CO space, if X is compact and every closed subspace of X is homeomorphic to a clopen subspace of X, and X is a hereditarily CO space (HCO space), if every closed subspace is a CO space. It is well-known that every well-ordered chain with a last element, endowed with the interval topology, is an HCO space, and every HCO space is scattered. In this paper, we show the following theorems: Theorem (R. Bonnet): Every HCO space which is a continuous image of a compact totally disconnected interval space is homeomorphic to β+1 for some ordinal β. Every HCO space of countable Cantor-Bendixson rank is homeomorphic to α+1 for some countable ordinal α. Theorem (S. Shelah): Assume(Formula presented.). Then there is a HCO compact space X of Cantor-Bendixson rank ω1} and of cardinality ℵ1 such that: X has only countably many isolated points, Every closed subset of X is countable or co-countable, Every countable closed subspace of X is homeomorphic to a clopen subspace, and every uncountable closed subspace of X is homeomorphic to X, and X is retractive. In particular X is a thin-tall compact space of countable spread, and is not a continuous image of a compact totally disconnected interval space. The question whether it is consistent with ZFC, that every HCO space is homeomorphic to an ordinal, is open.
UR - http://www.scopus.com/inward/record.url?scp=51249170034&partnerID=8YFLogxK
U2 - 10.1007/BF02760945
DO - 10.1007/BF02760945
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AN - SCOPUS:51249170034
SN - 0021-2172
VL - 84
SP - 289
EP - 332
JO - Israel Journal of Mathematics
JF - Israel Journal of Mathematics
IS - 3
ER -