Infinitary operations, such as products indexed by countably infinite linear orders, arise naturally in the context of fundamental groups and groupoids. Despite the fact that the usual binary operation of the fundamental group determines the operation of the fundamental groupoid, we show that, for a locally path-connected metric space, the well-definedness of countable dense products in the fundamental group need not imply the well-definedness of countable dense products in the fundamental groupoid. Additionally, we show the fundamental groupoid \(\Pi _1(X)\) has well-defined dense products if and only if X admits a generalized universal covering space.

中文翻译：

---

基本类群中的致密产品

在基本组和类群环境中，自然产生无穷运算，例如以无穷无穷的线性阶数索引的产品。尽管基本组的通常二元运算决定了基本组群的运算，但我们表明，对于局部路径连接的度量空间，基本组中可数稠密乘积的良好定义并不一定意味着基群中可数稠密产物的定义。此外，我们证明，只有且仅当X承认广义的通用覆盖空间时，基类群\（\ Pi _1（X）\）才具有定义明确的密集乘积。