We prove that if X is a regular space with no uncountable free sequences, then the tightness of its \(G_\delta \) topology is at most the continuum and if X is, in addition, assumed to be Lindelöf then its \(G_\delta \) topology contains no free sequences of length larger then the continuum. We also show that, surprisingly, the higher cardinal generalization of our theorem does not hold, by constructing a regular space with no free sequences of length larger than \(\omega _1\), but whose \(G_\delta \) topology can have arbitrarily large tightness.

中文翻译：

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$$ G_ \ delta $$ Gδ-拓扑的紧密度的上限

我们证明如果X是没有不可数自由序列的规则空间，则其\（G_ \ delta \）拓扑的紧密度最多是连续体，并且如果X另外被假定为Lindelöf，则其\（G_ \ delta \）拓扑不包含长度大于连续统的自由序列。我们还表明，令人惊讶的是，通过构造不包含长度大于\（\ omega _1 \）的自由序列但其\（G_ \ delta \）拓扑可以构成的自由序列的规则空间，我们的定理的高阶基本概括不成立。具有任意大的密封性。