A space homeomorphic to each uncountable closed subspace under CH (Q1315473)

M. Rubin asked which compact Hausdorff spaces are homeomorphic to each of their uncountable closed subspaces. It is easily seen that such spaces are scattered and that the ordinal space \(\omega_ 1+1\) and the one- point compactification of a discrete space of cardinality \(\omega_ 1\) are two such examples. The author reports that Gruenhage has shown that it follows from PFA that the above two are the only examples while Shelah has constructed an Ostaszewski-type example from \(\diamondsuit\). This paper is an interesting example of making do with CH when one might reasonably expect that someting like \(\diamondsuit\) is required. The technique, very roughly, is to apply the ``Kunen-line'' technique to a Luzin set of reals. The resulting space is the one-point compactification of a thin-tall, locally compact, locally countable scattered space. The construction is, naturally, a rather involved induction.