A note on η_1-spaces - Bennett, Harold; Davis, Sheldon; Lutzer, David (Topology Proceedings, 2020)
In this paper we study some topological properties of η_1-spaces, i.e., topological spaces that use the open-interval topology of theη_1-sets that were introduced by Hausdorff more than a century ago. We focus on paracompactness, normality of products, topological completeness of various kinds, and certain generalized metric properties such as the existence of a small diagonal. In many cases, we find an intimate relation between topological properties of small η_1-spaces (i.e. having cardinality 2^ω) and the Continuum Hypothesis (CH). For example, we show that (CH) is equivalent to the statement that if X is an η_1-space of cardinality 2^ω, then X_n is hereditarily paracompact and monotonically normal and is homeomorphic to X for every finite n≥ 1, and we show that CH is equivalent to the statement that every η_1-space of cardinality 2^ω is realcompact. In addition, we investigate the role of Hušek’s small diagonal property, showing that an η_1-space X has a small diagonal if and only if each subset SX with |S|≤ω_1 is closed. Consequently, under CH, no η_1-space with cardinality 2^ω can have a small diagonal, and we show that that if CH fails, then is is undecidable whether each ultrapower R^ωU must have a small diagonal. Under CH, we show that any finite power of any GO-modification of a small η_1-set is both monotonically normal and paracompact, and is homeomorphic to its square. We pose several questions about the topology of small η_1-spaces in models where the Continuum Hypothesis fails.