CH，Rolewicz问题和双离散系统
CH, a problem of Rolewicz and bidiscrete systems
摘要：
本文在$CH$下构造了一个不可度量化的紧T2空间$K$，使得在$C(K)$中的不可数半双正交序列都是一种非常特殊的序列.空间$K$具有许多优良的性质，如遗传可分、遗传Lindelöf和度量空间的2：1连续原像，以及$K$上的所有Radon测度都是可分的。然而，$K$不是Rosenthal紧。我们在紧空间中引入了双离散系统的概念，并指出每个无限紧T2空间$K$必须有一个大小为$d(K)$的双离散系统，即$K$的密度。这特别意味着$C(K)$具有大小为$d(K)$的双正交系。
英文摘要：
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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