LemmaStatement: Literature-sourcedProof: AI-generatedprecheck passaudited 2026-09-01

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The kernels of the finite coordinate projections form an open normal neighbourhood basis at the identity

Statement

For a profinite group presented as an inverse limit of finite discrete groups, the kernels of the coordinate projections form an open normal neighbourhood basis at the identity.

Facts & Assumptions

Given: A profinite group written as $L = \underset{\leftarrow}{l i m} G_{i}$ with coordinate maps $π_{i} : L \to G_{i}$ .

[L1]

Each $π_{i}$ is a coordinate projection from the inverse limit (The inverse limit has its canonical coordinate projection maps).

[F1]

Open sets in $L$ come from cylinders in the product topology, and a profinite group is such an inverse limit (The inverse limit of finite groups carries the subspace topology from the product of discrete factors, A profinite group is a topological group isomorphic to an inverse limit of finite discrete groups, Normal subgroup: invariance under conjugation, The quotient group $G / N$ and coset product $(g N) (h N) = g h N$ ).

Proof

technique · direct

1.1

Each kernel $\ker π_{i}$ is a normal subgroup because $π_{i}$ is a homomorphism, and it is open because ${e_{i}} \subseteq G_{i}$ is open in the discrete factor and $π_{i}^{- 1} ({e_{i}}) = \ker π_{i}$ .

L1F1given

1.2

Let $U$ be any neighbourhood of the identity in $L$ . By [F1], some basic cylinder fixing finitely many coordinates, say those in a finite set $F \subseteq I$ , is contained in $U$ . Choose $k \in I$ dominating every index in $F$ by directedness. If $x \in \ker π_{k}$ , then compatibility gives $π_{i} (x) = φ_{i k} (e_{k}) = e_{i}$ for every $i \in F$ , so $x$ lies in that cylinder. Hence $\ker π_{k} \subseteq U$ .

F1L1givenchoose

2.1

Therefore the family $(\ker π_{i})$ is a neighbourhood basis of open normal subgroups at the identity.

step 1.1step 1.2∎

Depends on

Used by

Assuming Choice, a topological group is profinite exactly when it is compact, Hausdorff, and totally disconnected Theorem

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

Brian Osserman, Math 6112 notes on inverse limits and profinite groups (standard reference, not scraped)
H. W. Lenstra, Profinite groups and Galois groups (standard reference, not scraped)