Infinite product operations are at the forefront of the study of homotopy groups of Peano continua and other locally path-connected spaces. In this paper, we define what it means for a space X to have infinitely commutative $\pi _1$ -operations at a point $x\in X$ . Using a characterization in terms of the Specker group, we identify several natural situations in which this property arises. Maintaining a topological viewpoint, we define the transfinite abelianization of a fundamental group at any set of points $A\subseteq X$ in a way that refines and extends previous work on the subject.

中文翻译：

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基本群中的无限交换性和阿贝化

无限乘积操作处于研究 Peano continua 和其他局部路径连接空间的同伦群的前沿。在本文中，我们定义了空间X在点 $x\in X$ 处具有无限可交换 $\pi _1$ -操作的含义。使用 Specker 组的特征，我们确定了出现该属性的几种自然情况。保持拓扑观点，我们在任何点集 $A\subseteq X$ 处定义基本群的超限阿贝尔化，以改进和扩展先前关于该主题的工作。