关于稠密完全度量空间和uniformly continuous函数在$\mathbf{ZF}$中的扩张
On densely complete metric spaces and extensions of uniformly continuous functions in $\mathbf{ZF}$
摘要：
度量空间$\mathbf{X}$称为稠密完备的，如果在$\mathbf{X}$中存在稠密集$D$，使得$D $的每个柯西点列收敛于$\mathbf{X}$。本文的主要目的之一是证明可数选择公理（缩写为$\mathbf{CAC}$）等价于下列陈述：\smallskip（i）每个稠密完备（连通）度量空间$\mathbf{X}$是完备的。\smallskip \（ii）对于每对度量空间$\mathbf{X}$和$\mathbf{Y}$，如果$% \mathbf{Y}$是完备的，$\mathbf{S}$是$\mathbf{X}$ %的稠密子空间，而$f:\mathbf{S}\rightarrow \mathbf{Y}$是一致连续函数，则存在$f$的一致连续扩张$F:\mathbf{X}\to% \mathbf{Y}$. \smallskip（iii）度量空间的完备子空间有完备闭包。\smallskip（iv）度量空间的完备子空间是闭的。还证明了（i）对实直线的子集的限制等价于$\mathbf{CAC}$对$\mathbb{R}$的子集的限制$\mathbf{CAC}(\mathbb{R})$。然而，（ii）对$% \mathbb{R}$子集的限制严格弱于$\mathbf{CAC}(\mathbb{R})$，因为它与$\mathbb{R}$是顺序的陈述等价。此外，在其他相关的结果，证明了，对任何正整数$% n$，空间$\mathbb{R}^n$是序列的当且仅当$\mathbb{R}$是序列的。证明了$\mathbb{R}\times\mathbb{Q}$不是稠密完备的当且仅当$\mathbf{CAC}(\mathbb{R})$成立.
英文摘要：
A metric space $\mathbf{X}$ is called densely complete if there exists a dense set $D$ in $\mathbf{X}$ such that every Cauchy sequence of points of $D $ converges in $\mathbf{X}$. One of the main aims of this work is to prove that the countable axiom of choice, $\mathbf{CAC}$ for abbreviation, is equivalent with the following statements:\smallskip (i) Every densely complete (connected) metric space $\mathbf{X}$ is complete.\smallskip\ (ii) For every pair of metric spaces $\mathbf{X}$ and $\mathbf{Y}$, if $% \mathbf{Y}$ is complete and $\mathbf{S}$ is a dense subspace of $\mathbf{X}$% , while $f:\mathbf{S}\rightarrow \mathbf{Y}$ is a uniformly continuous function, then there exists a uniformly continuous extension $F:\mathbf{X}\to% \mathbf{Y}$ of $f$.\smallskip (iii) Complete subspaces of metric spaces have complete closures.\smallskip (iv) Complete subspaces of metric spaces are closed.\smallskip It is also shown that the restriction of (i) to subsets of the real line is equivalent to the restriction $\mathbf{CAC}(\mathbb{R})$ of $\mathbf{CAC}$ to subsets of $\mathbb{R}$. However, the restriction of (ii) to subsets of $% \mathbb{R}$ is strictly weaker than $\mathbf{CAC}(\mathbb{R})$ because it is equivalent with the statement that $\mathbb{R}$ is sequential. Moreover, among other relevant results, it is proved that, for every positive integer $% n$, the space $\mathbb{R}^n$ is sequential if and only if $\mathbb{R}$ is sequential. It is also shown that $\mathbb{R}\times\mathbb{Q}$ is not densely complete if and only if $\mathbf{CAC}(\mathbb{R})$ holds.
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