基本群胚中的超限乘积约化
Transfinite product reduction in fundamental groupoids
AI总结：
本文证明了基本群胚中超限乘积约化的技术引理，并据此得出：若空间 $X$ 的代数 $1$-野集分散且超限基本群运算良定义，则所有超限基本群胚运算亦良定义。
摘要：
在基本群胚的背景下，由可数无限线性序索引的无限乘积自然出现。如果索引序允许包含一个稠密子序，则此类乘积被称为“超限”的，否则被称为“分散”的。本文证明了与基本群胚中超限乘积的约化（即因子的合并）相关的几个技术引理。应用这些结果，我们证明：如果空间 $X$ 具有分散的代数 $1$-野集 $\text{aw}(X)$ 且超限基本群运算良定义，则所有超限基本群胚运算也是良定义的。
英文摘要：
Infinite products, indexed by countably infinite linear orders, arise naturally in the context of fundamental groupoids. Such products are called "transfinite" if the index orders are permitted to contain a dense suborder and are called "scattered" otherwise. In this paper, we prove several technical lemmas related to the reduction (i.e. combining of factors) of transfinite products in fundamental groupoids. Applying these results, we show that if the transfinite fundamental group operations are well-defined in a space $X$ with a scattered algebraic $1$-wild set $\mathbf{aw}(X)$, then all transfinite fundamental groupoid operations are also well-defined.
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