We investigate C-sets in almost zero-dimensional spaces, showing that closed $\sigma$C-sets are C-sets. As corollaries, we prove that every rim-$\sigma$-compact almost zero-dimensional space is zero-dimensional and that each cohesive almost zero-dimensional space is nowhere rational. To show these results are sharp, we construct a rim-discrete connected set with an explosion point. We also show every cohesive almost zero-dimensional subspace of $($Cantor set$)$$\times\mathbb R$ is nowhere dense.

中文翻译：

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在有凝聚力的几乎零维空间上

我们研究了几乎零维空间中的 C-sets，表明封闭的 $\sigma$C-sets 是 C-sets。作为推论，我们证明了每个 rim-$\sigma$-compact 几乎零维空间都是零维的，并且每个有凝聚力的几乎零维空间都是无理的。为了显示这些结果是清晰的，我们构建了一个带有爆炸点的边缘离散连接集。我们还展示了 $($Cantor set$)$$\times\mathbb R$ 的每个有凝聚力的几乎零维子空间都不稠密。