分裂区间的选择性质与连续统假设
Selection properties of the split interval and the Continuum Hypothesis
摘要：
证明了从可度量化可分空间$X$到GO-空间$Y$的每个usco多重映射$Φ:X\to Y$都有$F_σ$ -可测选择.另一方面，对于分割区间$\ddot{\mathbb I}$和它的平方在单位平方${\mathbb I}^2$上的投影$P:\ddot{\mathbb I}^2\to{\mathbb I}^2$，usco多重映射$P^{-1}:{\mathbb I}^2\multimap\ddot{\mathbb I}^2$具有Borel（$F_σ$ -可测量的）选择当且仅当连续统假设成立。这个CH-例子表明，已知的结果Borel选择usco映射到可分割的紧空间不能扩展到更广泛的一类紧空间。
英文摘要：
We prove that every usco multimap $Φ:X\to Y$ from a metrizable separable space $X$ to a GO-space $Y$ has an $F_σ$-measurable selection. On the other hand, for the split interval $\ddot{\mathbb I}$ and the projection $P:\ddot{\mathbb I}^2\to{\mathbb I}^2$ of its square onto the unit square ${\mathbb I}^2$, the usco multimap $P^{-1}:{\mathbb I}^2\multimap\ddot{\mathbb I}^2$ has a Borel ($F_σ$-measurable) selection if and only if the Continuum Hypothesis holds. This CH-example shows that know results on Borel selections of usco maps into fragmentable compact spaces cannot be extended to a wider class of compact spaces.
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