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 the$η_1$-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 $S\subseteq X$ 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 $\mathbb R^ω\setminus \mathcal U$ 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.