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The inverse-limit topology on Zp agrees with the p-adic metric topology

Statement

The inverse-limit topology on $Z_{p}$ coincides with the topology induced by the metric $d_{p}$ .

Facts & Assumptions

Given: An element $x = (x_{n})$ of $Z_{p}$ and an integer $n \geq 1$ .

[F1]

The metric on $Z_{p}$ is defined by the largest initial block of equal residue coordinates (The p-adic metric on Zp is determined by the first coordinate at which two compatible residue systems differ).

[L1]

The inverse-limit topology is the subspace topology from the product of the discrete quotients, and cylinder traces form a basis (The inverse limit of finite groups carries the subspace topology from the product of discrete factors).

[L2]

An element of $Z_{p}$ is exactly a compatible residue-class tuple (The p-adic integers are the compatible residue-class tuples in the inverse limit of Z mod p^n).

Proof

technique · direct

1.1

Let $U_{n} (x) : = {y = (y_{r}) \in Z_{p} : y_{r} = x_{r} for 1 \leq r \leq n}$ . By [F1], this is exactly the metric ball $U_{n} (x) = {y \in Z_{p} : d_{p} (x, y) \leq p^{- n}},$ because the first $n$ residue coordinates agree if and only if the largest initial block of equal coordinates has length at least $n$ .

F1L2givenalgebra

2.1

By [L1], the same set $U_{n} (x)$ is the trace on $Z_{p}$ of the cylinder in the product space that fixes the first $n$ coordinates, so every basic metric ball is inverse-limit open. Conversely, let a basic inverse-limit cylinder $C$ contain $x$ and restrict the finite set of coordinates $F$ . Taking $n \geq r$ for every $r \in F$ , compatibility in [L2] shows that $U_{n} (x) \subseteq C$ . Thus the sets $U_{n} (x)$ refine every inverse-limit neighbourhood of $x$ .

L1L2step 1.1algebra

3.1

The sets $U_{n} (x)$ therefore form a neighbourhood basis for both topologies at every point $x$ . So the inverse-limit topology and the metric topology coincide.

step 1.1step 2.1∎

Depends on

Used by

Dependency tree · two levels

7 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Gareth Wilkes, Profinite Groups and Group Cohomology lecture notes (standard reference, not scraped)
Jordan Bell, The profinite completion of the integers, the p-adic integers, and Prufer p-groups (standard reference, not scraped)