可数集的几乎不交族与可分补性质
Almost disjoint families of countable sets and separable complementation properties
摘要：
研究了由不可数集的可数子集的几乎不相交族${\mathcal A}$诱导的$K_{\mathcal A}$上连续函数的可分补性及其在巴拿赫空间中的自然变化.对于这些空间，我们证明了$C(K_{\mathcal A})$具有可分互补性质的受控变体当且仅当$C(K_{\mathcal A})$在弱拓扑中是Lindelöf当且仅当$K_{\mathcal A}$是monolithic。我们给出了一个${\mathcal A}$的例子，其中$C(K_{\mathcal A})$具有SCP，而$K_{\mathcal A}$不是整体的，并且是一个具有受控和连续SCP的空间$C(K_{\mathcal A})$的例子，它既没有投影骨架，也没有恒等式的投影分解。最后，我们描述的结构几乎不相交的家庭的基数$ω_1$诱导单片空间的形式$K_{\mathcal A}$：它们可以从可数许多阶梯系统和成对不相交的家庭，应用简单的操作。
英文摘要：
We study the separable complementation property (SCP) and its natural variations in Banach spaces of continuous functions over compacta $K_{\mathcal A}$ induced by almost disjoint families ${\mathcal A}$ of countable subsets of uncountable sets. For these spaces, we prove among others that $C(K_{\mathcal A})$ has the controlled variant of the separable complementation property if and only if $C(K_{\mathcal A})$ is Lindelöf in the weak topology if and only if $K_{\mathcal A}$ is monolithic. We give an example of ${\mathcal A}$ for which $C(K_{\mathcal A})$ has the SCP, while $K_{\mathcal A}$ is not monolithic and an example of a space $C(K_{\mathcal A})$ with controlled and continuous SCP which has neither a projectional skeleton nor a projectional resolution of the identity. Finally, we describe the structure of almost disjoint families of cardinality $ω_1$ which induce monolithic spaces of the form $K_{\mathcal A}$: They can be obtained from countably many ladder systems and pairwise disjoint families applying simple operations.
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