拓扑范畴上的拟序
Quasiorders on topological categories
摘要：
本文证明了，对每一个基数$α\geq {\mathfrak c}$，存在一个具有$|X|=α$的可度量化空间$X$，使得对于集合$Q$上的每一对拟序$\leq_1$，$\leq_2$，$|Q| \leq α$满足蕴涵$$q \leq_1 q' \implies q \leq_2 q'$$，存在一个由$X$的非同胚闭子集组成的系统$\{X(q) : q\in Q\}$，它具有以下性质：（1）$q \leq_1 q'$当且仅当$X(q)$同胚于$X(q')$的闭序子集，（2）$q \leq_2 q'$意味着$X(q)$同胚于$X(q')$的闭子集;（3）$\neg (q \leq_2 q')$意味着不存在一对一$X(q)$到$X(q')$的一个连续映射。
英文摘要：
We prove that, for every cardinal number $α\geq {\mathfrak c}$, there exists a metrizable space $X$ with $|X|=α$ such that for every pair of quasiorders $\leq_1$, $\leq_2$ on a set $Q$ with $|Q| \leq α$ satisfying the implication $$q \leq_1 q' \implies q \leq_2 q'$$ there exists a system $\{X(q) : q\in Q\}$ of non-homeomorphic clopen subsets of $X$ with the following properties: (1) $q \leq_1 q'$ if and only if $X(q)$ is homeomorphic to a clopen subset of $X(q')$, (2) $q \leq_2 q'$ implies that $X(q)$ is homeomorphic to a closed subset of $X(q')$ and (3) $\neg (q \leq_2 q')$ implies that there is no one-to-one continuous map of $X(q)$ into $X(q')$.
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