Tukey order is used to compare the cofinal complexity of partially order sets (posets). We prove that there is a $2^{\mathfrak{c}}$-sized collection of sub-posets in $2^{\omega }$ which forms an antichain in the sense of Tukey ordering. Using the fact that any boundedly-complete sub-poset of ${\omega }^{\omega }$ is a Tukey quotient of ${\omega }^{\omega }$, we answer two open questions published in [14].

The relation between P-base and strong Pytkeev^⁎ property is investigated. Let P be a poset equipped with a second-countable topology in which every convergent sequence is bounded. Then we prove that any topological space with a P-base has the strong Pytkeev^⁎ property. Furthermore, we prove that every uncountably-dimensional locally convex space (lcs) with a P-base contains an infinite-dimensional metrizable compact subspace. Examples in function spaces are given.

Tukey 阶用于比较偏序集（poseets）的共同最终复杂度。我们证明有一个$2^{\mathfrak{C}}$-大小的子姿势集合 $2^{\omega }$它形成了 Tukey 排序意义上的反链。使用任何有界完全子集的事实${\omega }^{\omega }$ 是 Tukey 商 ${\omega }^{\omega }$，我们回答了 [14] 中发表的两个开放性问题。

之间的关系P碱基强Pytkeev ^⁎性能进行了研究。设P是一个配备第二可数拓扑的偏序集，其中每个收敛序列都是有界的。然后，我们证明了任何拓扑空间P碱基具有较强的Pytkeev ^⁎财产。此外，我们证明了每个具有P基的不可数维局部凸空间（lcs）都包含一个无限维可度量的紧子空间。给出了函数空间中的例子。