We examine the Borel version of the \(\sigma \)-finite chain condition of Horn and Tarski for a class of posets T(X) which have been used in the solution of their well-known problem. More precisely, we show that the poset \(T(\pi \mathbb{Q} \it )\) does not have the \(\sigma \)-finite chain condition witnessed by Borel pieces. More precisely, we define a condition on the topological spaces X under which the corresponding Todorcevic ordering T(X) does not have the \(\sigma \)-bounded chain condition witnessed by a countable Borel decomposition although it might satisfy the \(\sigma \)-finite chain condition witnessed by a non Borel decomposition.

中文翻译：

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T(X) 的 Borel 链条件

我们检查了 Horn 和 Tarski 的 \(\sigma \)-有限链条件的 Borel 版本，用于解决他们众所周知的问题的一类偏序集 T(X)。更准确地说，我们证明了偏序集 \(T(\pi \mathbb{Q} \it )\) 不具有由 Borel 碎片见证的 \(\sigma \)-有限链条件。更准确地说，我们在拓扑空间 X 上定义了一个条件，在该条件下，相应的 Toodorcevic 排序 T(X) 不具有可数 Borel 分解所见证的 \(\sigma \)-有界链条件，尽管它可能满足 \(\ sigma \)-非 Borel 分解所见证的有限链条件。