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

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A map into an inverse limit is continuous exactly when all coordinate composites are continuous

Statement

A map into an inverse limit of finite discrete groups is continuous if and only if all of its coordinate composites are continuous.

Facts & Assumptions

Given: A map $f : Y \to L = \underset{\leftarrow}{l i m} G_{i}$ , where $L$ has its inverse-limit topology.

[L1]

The coordinate projections $π_{i} : L \to G_{i}$ are defined on the inverse limit (The inverse limit has its canonical coordinate projection maps).

[F1]

The inverse-limit topology is the subspace topology from the product, and a map into a product is continuous exactly when every coordinate composite is continuous (The inverse limit of finite groups carries the subspace topology from the product of discrete factors, Continuity of a map of topological spaces at a point and globally).

Proof

technique · direct

1.1

If $f$ is continuous, then every composite $π_{i} \circ f$ is continuous because it is the composite of two continuous maps, namely $f$ and the coordinate map [L1].

L1F1given

1.2

Conversely, suppose every $π_{i} \circ f$ is continuous. Regard $f$ as a map into the ambient product. By the product-topology clause of [F1], those coordinatewise continuities imply that the product-valued map is continuous. Since the image already lies in the subspace $L$ , the same map is continuous as a map into $L$ .

F1L1given

2.1

Steps 1.1 and 1.2 prove the equivalence.

step 1.1step 1.2∎

Depends on

Used by

Dependency tree · two levels

9 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)