The combinatorics of sub-Ostaszewski spaces (Q2701859)

The author gives a condition which is equivalent to the existence of a sub-Ostaszewski space. \(X\subset 2^{\omega_1}\) is an HDT if every infinite subset of \(X\) is finally dense. \(X\) is a weak HDT, if every \(Y\in [X]^{\omega_1}\) has an infinite finally dense subset. Then, ``there is a sub-Ostaszewski space'' and ``there is a weak HDT pre-algebra \(J\subset [\Omega_1]^{\omega}\)'' are equivalent. For example, \(J\) is defined as \(\{u \subset X : u\) is countable and clopen\}. Then \(J\) has the properties required.NEWLINENEWLINENEWLINEMoreover, she proves two propositions: There is a locally compact sub-Ostaszewski space and there is a weak HDT pre-algebra \(J\subset [\omega_1]^\omega\) that no \(K\in J\) is the union of a chain \(C\) of elements of \(J\) with \(K\notin C\) are equivalent.