Recent zbMATH articles in MSC 54Ghttps://zbmath.org/atom/cc/54G2021-06-15T18:09:00+00:00WerkzeugA note on \(\eta_1\)-spaces.https://zbmath.org/1460.540212021-06-15T18:09:00+00:00"Bennett, Harold"https://zbmath.org/authors/?q=ai:bennett.harold-r"Davis, Sheldon"https://zbmath.org/authors/?q=ai:davis.sheldon-w"Lutzer, David"https://zbmath.org/authors/?q=ai:lutzer.david-jFelix Hausdorff defined an \(\eta_1\)-set to be a linearly ordered set \((X, <)\) such that for any two countable subsets \(A, B\) with \(A < B\), the interval \((a,b)\) is non-empty for every \(a \in A, b \in B\). The authors study \(\eta_1\)-sets \(X\) with the open interval topology and with \(|X| = 2^\omega\), which they call small \(\eta_1\)-sets. In an extensive study of such sets, they include many nice results and examples addressing products, monotonically normality, and paracompactness, among other topics. The authors show that the Continuum Hypothesis is equivalent to each of the following: a) every small \(\eta_1\)-set is paracompact, b) every small \(\eta_1\)-set is realcompact, c) every small \(\eta_1\)-set \(X\) is homeomorphic to \(X^2\), and d) for every small \(\eta_1\)-set \(X\), \(X^2\) is normal. Several open questions are also presented.
Reviewer: Thomas Richmond (Bowling Green)