Tychonoff空间中射影集的约化与分离
On reduction and separation of projective sets in Tychonoff spaces
摘要：
证明了对任意的Tychonoff空间$X$和Hausdorff运算$\mathbfΦ$，由$X$中的零集通过$\mathbfΦ$生成的类$\mathbfΦ(\mathscr Z,X)$具有约化或分离性质，如果相应的实数集类$\mathbfΦ(\mathscr F,\mathbb R)$具有相同的性质.特别地，在投影确定性下，$X$中的这些投射集的性质与实数的投射集的第一周期性定理状态具有相同的模式：类$\mathbfΣ^{1}_{2n}(\mathscr Z,X)$和$\mathbfΠ^{1}_{2n+1}(\mathscr Z,X)$具有约化性质，而$\mathbfΠ^{1}_{2n}(\mathscr Z,X)$和$\mathbfΣ^{1}_{2n+1}(\mathscr Z,X)$具有分离性质。
英文摘要：
We show that for every Tychonoff space $X$ and Hausdorff operation $\mathbfΦ$, the class $\mathbfΦ(\mathscr Z,X)$ generated from zero sets in $X$ by $\mathbfΦ$ has the reduction or separation property if the corresponding class $\mathbfΦ(\mathscr F,\mathbb R)$ of sets of reals has the same property. In particular, under Projective Determinacy, these properties of such projective sets in $X$ have the same pattern as the First Periodicity Theorem states for projective sets of reals: the classes $\mathbfΣ^{1}_{2n}(\mathscr Z,X)$ and $\mathbfΠ^{1}_{2n+1}(\mathscr Z,X)$ have the reduction property while $\mathbfΠ^{1}_{2n}(\mathscr Z,X)$ and $\mathbfΣ^{1}_{2n+1}(\mathscr Z,X)$ have the separation property.
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