A note on $\eta_1$-spaces
Keywords:
eta1-set, eta1-space, paracompact, monotonically normal, Dieudonné complete, realcompact, A-metric space, Baire space, Cp(X), small diagonal, ultrapower, homeomorphic to square, Continuum HypothesisAbstract
In this paper we study some topological properties of $\eta_1$-spaces, i.e., topological spaces that use the open-interval topology
of the$\eta_1$-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 $\eta_1$-spaces (i.e. having cardinality $2^\omega$) and the Continuum Hypothesis (CH). For example, we show that (CH) is equivalent to the statement that if X is an $\eta_1$-space of cardinality $2^\omega$, then $X_n$ is hereditarily paracompact and monotonically normal and is homeomorphic to X for every finite $n\geq 1$, and we show that CH is equivalent to the statement that every $\eta_1$-space of cardinality $2^\omega$ is realcompact. In addition, we investigate the role of Hušek’s small diagonal property, showing that an $\eta_1$-space X has a small diagonal if and only if each subset $S\subseteq X$ with $|S|\leq\omega_1$ is closed. Consequently, under CH, no $\eta_1$-space with cardinality $2^\omega$ can have a small diagonal, and we show that that if CH fails, then is is undecidable whether each ultrapower $\mathbb R^\omega\setminus \mathcal U$ must have a small diagonal. Under CH, we show that any finite power of any GO-modification of
a small $\eta_1$-set is both monotonically normal and paracompact, and is homeomorphic to its square. We pose several questions about the topology of small $\eta_1$-spaces in models where the Continuum Hypothesis fails.
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