ON HCO SPACES - AN UNCOUNTABLE COMPACT T-2 SPACE, DIFFERENT FROM ALEPH(1)+1, WHICH IS HOMEOMORPHIC TO EACH OF ITS UNCOUNTABLE CLOSED SUBSPACES

Let X be an Hausdorff space. We say that X is a CO space, if X is comp act and every closed subspace of X is homeomorphic to a clopen subspac e 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 sp ace, and every HCO space is scattered. In this paper, we show the foll owing theorems: THEOREM (R. Bonnet): (a) Every HCO space which is a co ntinuous image of a compact totally disconnected interval space is hom eomorphic to beta+1 for some ordinal beta. (b) Every HCO space of coun table Cantor-Bendiuson rank is homeomorphic to alpha + 1 for some coun table ordinal alpha. THEOREM (S. Shelah): Assume lozenge N-1. Then the re is a HCO compact space X of Cantor-Bendixson rank w(1) and of cardi nality N-1 such that: (1) X has only countably many isolated points, ( 2) Every closed subset of X is countable or co-countable, (3) Every co untable closed subspace of X is homeomorphic to a clopen subspace, and every uncountable closed subspace of X is homeomorphic to X, and (4) X is retractive. In particular X is a thin-tall compact space of count able spread, and is not a continuous image of a compact totally discon nected interval space. The question whether it is consistent with ZFC, that every HCO space is homeomorphic to an ordinal, is open.