A space has $\sigma$-compact tightness if the closures of $\sigma$-compact subsets determines the topology. We consider a dense set variant that we call densely k-separable. We consider the question of whether every densely k-separable space is separable. The somewhat surprising answer is that this property, for compact spaces, implies that every dense set is separable. The path to this result relies on the known connections established between $\pi$-weight and the density of all dense subsets, or more precisely, the cardinal invariant $\delta(X)$.

中文翻译：

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稠密的 k 可分离致密体是稠密可分离的

如果 $\sigma$-compact 子集的闭包决定了拓扑，则空间具有 $\sigma$-compact 紧密度。我们考虑一个密集集变体，我们称之为密集 k 可分离。我们考虑是否每个密集的 k 可分空间都是可分的。有点令人惊讶的答案是，对于紧致空间，这个性质意味着每个稠密集都是可分的。这个结果的路径依赖于在 $\pi$-weight 和所有密集子集的密度之间建立的已知联系，或者更准确地说，是基数不变量 $\delta(X)$。