论文标题
$ \ mathbf {zf} $中的（quasi）-metrizable空间的CUF产品和CUF总和
Cuf products and cuf sums of (quasi)-metrizable spaces in $\mathbf{ZF}$
论文作者
论文摘要
CUF空间（set，分别）是一个空间（set，分别），它是有限子空间（子集，分别）的可数符合的结合。 It is proved in $\mathbf{ZF}$ (with the absence of the axiom of choice) that all countable unions of cuf (denumerable, resp.) sets are cuf sets iff all countable products of cofinite cuf (denumerable, resp.) spaces are quasi-metrizable iff all countable products of one-point Hausdorff compactifications of infinite cuf (denumerable,分别。如果FF是准分，则可以算出不可离散空间的单点Hausdorff压缩的可数产物。显示了$ \ Mathbf {Zf} $的模型，其中可计数的产品两点hausdorff的不可离散空间的hausdorff压缩是首先计算的，而无需准量化。还证明了其他相关的独立结果。
A cuf space (set, resp.) is a space (set, resp.) which is a countable union of finite subspaces (subsets, resp.). It is proved in $\mathbf{ZF}$ (with the absence of the axiom of choice) that all countable unions of cuf (denumerable, resp.) sets are cuf sets iff all countable products of cofinite cuf (denumerable, resp.) spaces are quasi-metrizable iff all countable products of one-point Hausdorff compactifications of infinite cuf (denumerable, resp.) spaces are quasi-metrizable. A countable product of one-point Hausdorff compactifications of denumerable discrete spaces is first-countable iff it is quasi-metrizable. A model of $\mathbf{ZF}$ is shown in which a countable product two-point Hausdorff compactifications of denumerable discrete spaces is first-countable without being quasi-metrizable. Other relevant independence results are also proved.
