紧的最终不同族
Tight Eventually Different Families
AI总结：
本文推广紧几乎不交族，定义紧最终不同函数族与排列集，证明其存在性、力迫保持及描述集合论性质，并据此计算多种模型中的 a_e 与 a_p。
摘要：
我们将紧的几乎不交族的概念加以推广，引入了 Baire 空间中函数的紧的最终不同族，以及 ω 的排列的紧的最终不同集合这两个概念。这类集合强化了极大性，在 MA(σ-linked) 下存在，并配有一个恰当性保持定理。紧性这一概念还推广了此前关于函数族极大性的力迫不可破坏性的工作。因此，我们通过给出显式见证，在许多已知模型中计算了基数 a_e 与 a_p，从而得到连续统基数特征若干构型的协调性，包括 a_e = a_p = d < a_T、a_e = a_p < d = a_T、a_e = a_p = u < non(N) = cof(N)，以及 a_e = a_p = i < u。我们还证明在 L 中存在 Π^1_1 的紧最终不同族和紧最终不同排列集合，从而在得到上述不等式的同时，获得用于 a_e = a_p = aleph_1 的 Π^1_1 见证。此外，我们证明紧最终不同族是 Cohen 不可破坏的，并且绝不可能是解析的。
英文摘要：
Generalizing the notion of a tight almost disjoint family, we introduce the notions of a {\em tight eventually different} family of functions in Baire space and a {\em tight eventually different set of permutations} of $ω$. Such sets strengthen maximality, exist under $\mathsf{MA} (σ{\rm -linked})$ and come with a properness preservation theorem. The notion of tightness also generalizes earlier work on the forcing indestructibility of maximality of families of functions. As a result we compute the cardinals $\mathfrak{a}_e$ and $\mathfrak{a}_p$ in many known models by giving explicit witnesses and therefore obtain the consistency of several constellations of cardinal characteristics of the continuum including $\mathfrak{a}_e = \mathfrak{a}_p = \mathfrak{d} < \mathfrak{a}_T$, $\mathfrak{a}_e = \mathfrak{a}_p < \mathfrak{d} = \mathfrak{a}_T$, $\mathfrak{a}_e = \mathfrak{a}_p = \mathfrak{u} < non(\mathcal N) = cof(\mathcal N)$ and $\mathfrak{a}_e = \mathfrak{a}_p =\mathfrak{i} < \mathfrak{u}$. We also show that there are $Π^1_1$ tight eventually different families and tight eventually different sets of permutations in $L$ thus obtaining the above inequalities alongside $Π^1_1$ witnesses for $\mathfrak{a}_e = \mathfrak{a}_p = \aleph_1$. Moreover, we prove that tight eventually different families are Cohen indestructible and are never analytic.
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