AI 中文总结
本文基于软集的软元素观点构建选择空间，从纤维度量定义规范实值度量，比较诱导拓扑等，给出多种收缩映射的不动点结果，还包括稳定性等估计，主要贡献是针对不可数参数集，区分不同收敛性并阐明收缩假设。
AI 中文摘要
软集\((F,E)\)为每个参数\(e\in E\)分配全集\(X\)的一个子集\(F(e)\)。从软元素角度看，点是一个选择\(x:E\to X\)且对所有参数\(x(e)\in F(e)\)，从而\((F,E)\)产生具体选择空间\(\SE(F)=\prod_{e\in E}F(e)\)。本文在此选择空间上建立了一个拓扑敏感的不动点框架。从纤维度量\((d_e)_{e\in E}\)出发，定义了两个规范实值度量：可数参数集的乘积度量\(\dPi\)和有界化后任意参数集的一致度量\(\dsup\)。比较了诱导拓扑，刻画了收敛性和完备性，给出了多种收缩映射的不动点结果，还包括稳定性和数据依赖性估计。主要拓扑贡献涉及不可数参数集：\(\SE(F)\)上的乘积拓扑通常不可度量化，而\(\dsup\)总是给出可度量化的一致拓扑。为处理不可度量化情况，引入由坐标伪度量生成的自然乘积一致性，并证明了一个参数化收缩定理，在乘积拓扑中产生唯一不动点且收敛。这区分了逐点/乘积收敛与一致收敛，并阐明了每种情况下所需的收缩假设。
英文摘要
A soft set $(F,E)$ determines the selection space $\SE(F)=\prod_{e\in E}F(e)$. This paper studies two natural structures on that space. For at most countable $E$, the series metric $\dPi$ induces the product topology. For arbitrary $E$, the sup metric $\dsup$ induces uniform convergence. We prove that these metric spaces are complete exactly when all fibres are complete. We then compare global contractions with coordinatewise contractions and give counterexamples when coordinate separability or a uniform contractive bound is absent. Standard fixed-point theorems for complete metric spaces are recorded as direct consequences, without repeating their classical proofs. For uncountable $E$, the product topology may fail to be metrizable, so we work with its product uniformity. A parameterwise contraction theorem gives a unique fixed point and convergence in the product topology; a common bound below one gives convergence in $\dsup$.
CommentsSubstantially revised after peer review. Elementary proofs of classical fixed-point theorems have been removed, the exposition has been shortened, assumptions and references have been corrected, and the treatment of uncountable parameter sets has been strengthened. 11 pages, 1 table
