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arXiv · 0910.3091

CH, a problem of Rolewicz and bidiscrete systems

MIrna Dzamonja·Istvan Juhasz
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Abstract

We give a construction under $CH$ of a non-metrizable compact Hausdorff space $K$ such that any uncountable semi-biorthogonal sequence in $C(K)$ must be of a very specific kind. The space $K$ has many nice properties, such as being hereditarily separable, hereditarily Lindelöf and a 2-to-1 continuous preimage of a metric space, and all Radon measures on $K$ are separable. However $K$ is not a Rosenthal compactum. We introduce the notion of bidiscrete systems in compact spaces and note that every infinite compact Hausdorff space $K$ must have a bidiscrete system of size $d(K)$, the density of $K$. This, in particular, implies that $C(K)$ has a biorthogonal system of size $d(K)$.

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MIrna Dzamonja, Istvan Juhasz. 2009-11-03. CH, a problem of Rolewicz and bidiscrete systems. https://arxiv.org/abs/0910.3091

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