We define a notion of tiling of the full infinite p-ary tree, establishing a series of equivalent criteria for a subtree to be a tile, each of a different nature; namely, geometric, algebraic, graph-theoretic, order-theoretic, and topological. We show how these results can be applied in a straightforward and constructive manner to define homeomorphisms between two given spaces of p-adic integers, \({\mathbb {Z}}_{p}\) and \({\mathbb {Z}}_{q}\), endowed with their corresponding standard non-archimedean metric topologies.

中文翻译：

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无限p-ary树和康托同胚的平铺

我们定义了一个完整的无限p 元树的平铺概念，为子树建立了一系列等效的标准，使其成为平铺，每个都有不同的性质；即几何、代数、图论、序论和拓扑。我们展示了如何以一种直接而有建设性的方式应用这些结果来定义两个给定的p进整数空间之间的同胚，\({\mathbb {Z}}_{p}\)和\({\mathbb {Z }}_{q}\)，赋予其相应的标准非阿基米德度量拓扑。