$ω^ω$中的子偏序集与强Pytkeev $^\ast$性质
Sub-posets in $ω^ω$ and the Strong Pytkeev$^\ast$ Property
摘要：
Tukey序被用来比较偏序集的共尾复杂度。我们证明了在$2^ω$中存在一个$2^\mathfrak{c}$大小的子集集合，它在Tukey序意义下构成一个反链.利用$ω^ω$的任意有界完备子集都是$ω^ω$的Tukey商这一事实，我们回答了发表在\cite{FKL16}上的两个公开问题.研究了$P$基与强Pytkeev $^\ast$性质的关系.设$P$是一个具有第二可数拓扑的偏序集，其中每个收敛序列都有界。证明了任何具有$P$ -基的拓扑空间都具有强Pytkeev $^\ast$性质.进一步证明了每一个具有$P$ -基的无限维局部凸空间（lcs）都包含一个无限维可度量化的紧致子空间.函数空间中的例子。
英文摘要：
Tukey order are 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 $2^ω$ which forms an antichain in the sense of Tukey ordering. Using the fact that any boundedly-complete sub-poset of $ω^ω$ is a Tukey quotient of $ω^ω$, we answer two open questions published in \cite{FKL16}. The relation between $P$-base and strong Pytkeev$^\ast$ 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$^\ast$ 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.
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