LemmaStatement: Literature-sourcedProof: AI-generatedprecheck passaudited 2026-09-04

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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Coordinatewise addition and negation make Zp a topological abelian group

Statement

If $x = (x_{n})$ and $y = (y_{n})$ are elements of $Z_{p}$ , then

$x + y : = (x_{n} + y_{n})_{n}, - x : = (- x_{n})_{n}$

are again elements of $Z_{p}$ . With these operations and the inverse-limit topology, $Z_{p}$ is a topological abelian group.

Facts & Assumptions

Given: Two elements $x = (x_{n})$ and $y = (y_{n})$ of $Z_{p}$ .

[F1]

An element of $Z_{p}$ is a compatible tuple $(x_{n})_{n \geq 1} \in \prod_{n} Z / p^{n} Z$ with $x_{n + 1} m o d p^{n} = x_{n}$ for every $n$ (The p-adic integers are the compatible residue-class tuples in the inverse limit of Z mod p^n).

Proof

technique · direct

1.1

Because reduction modulo $p^{n}$ is a group homomorphism, compatibility of the tuples in [F1] gives $(x_{n + 1} + y_{n + 1}) m o d p^{n} = x_{n} + y_{n}$ and $(- x_{n + 1}) m o d p^{n} = - x_{n}$ for every $n$ . Thus $x + y$ and $- x$ are again compatible tuples, so they lie in $Z_{p}$ .

F1givenalgebra

2.1

Associativity, commutativity, the zero element, and additive inverses all hold coordinatewise because they hold in every finite quotient $Z / p^{n} Z$ . The inverse-limit topology is the subspace topology from the product of the discrete coordinate groups, so coordinatewise addition and negation are continuous. Hence $Z_{p}$ is a topological abelian group.

F1step 1.1algebra∎

Depends on

The p-adic integers are the compatible residue-class tuples in the inverse limit of Z mod p^n

Used by

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

Dependency tree · two levels

3 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)