TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-09-04

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Zp is Hausdorff, totally disconnected, and complete, and compact assuming Choice

Statement

The space $Z_{p}$ is Hausdorff, totally disconnected, and complete for the $p$ -adic metric. Assuming the Axiom of Choice, it is also compact.

Facts & Assumptions

Given: A Cauchy sequence $(x^{(m)})_{m \geq 0}$ in $Z_{p}$ ; for the compactness clause, also the Axiom of Choice.

[L1]

An inverse limit of finite discrete groups is Hausdorff, compact, and totally disconnected (Inverse limits of finite discrete groups are Hausdorff and totally disconnected, and compact assuming Choice).

[L2]

The inverse-limit and $p$ -adic metric topologies on $Z_{p}$ agree (The inverse-limit topology on Zp agrees with the p-adic metric topology).

[L3]

Coordinatewise addition and negation make $Z_{p}$ a topological abelian group (Coordinatewise addition and negation make Zp a topological abelian group).

Proof

technique · direct

1.1

By construction, $Z_{p}$ is an inverse limit of the finite discrete groups $Z / p^{n} Z$ . Therefore [L1] gives that $Z_{p}$ is Hausdorff and totally disconnected, and also compact under the extra Choice hypothesis named in the Statement. The group structure from [L3] is already compatible with this topology.

L1L3givenalgebra

1.2

Fix $n \geq 1$ . Since the sequence is Cauchy and [L2] identifies the metric balls with the cylinder neighbourhoods, there exists $M_{n}$ such that for all $m, r \geq M_{n}$ the first $n$ coordinates of $x^{(m)}$ and $x^{(r)}$ agree. Let $a_{n}$ be that eventual common $n$ -th coordinate. The compatibility of the $x^{(m)}$ forces the tuple $a = (a_{n})_{n \geq 1}$ itself to be compatible, hence an element of $Z_{p}$ .

L2givenchooseconstruct

2.1

For each $n$ and every $m \geq M_{n}$ , the first $n$ coordinates of $x^{(m)}$ and $a$ agree, so step 1.2 gives $d_{p} (x^{(m)}, a) \leq p^{- n}$ . Given $ε > 0$ , choose $n$ with $p^{- n} < ε$ ; then every $m \geq M_{n}$ satisfies $d_{p} (x^{(m)}, a) < ε$ . Thus $x^{(m)} \to a$ in the $p$ -adic metric. Every Cauchy sequence therefore converges, so $Z_{p}$ is complete.

step 1.2L2algebra

3.1

Steps 1.1 and 2.1 prove the stated properties.

step 1.1step 2.1∎

Depends on

Used by

A standard p-adic digit sequence is Cauchy and converges in Zp Example

Dependency tree · two levels

13 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

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