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The inverse limit of finite groups carries the subspace topology from the product of discrete factors
Definition
If every in an inverse system is finite and discrete, the inverse limit carries the inverse-limit topology, meaning the subspace topology inherited from the product space where each factor has its discrete topology (The product set of functions choosing a point in each factor, the projections, the box topology, and the product topology as the initial topology of the projections; the empty product is a one-point space, Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace).
The intersections of with cylinder sets form a basis for this topology; an arbitrary open subset of is a union of such cylinder traces.
Depends on
- The inverse limit is the set of compatible tuples in the Cartesian product
- The product set $\prod_{i \in I} X_i$ of functions choosing a point in each factor, the projections, the box topology, and the product topology as the initial topology of the projections; the empty product is a one-point space
- Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace
Used by
- The profinite completion is the inverse limit of the finite quotients G over N Definition
- The inverse limit of finite discrete groups is a closed topological subgroup of the full product Lemma
- The kernels of the finite coordinate projections form an open normal neighbourhood basis at the identity Lemma
- A map into an inverse limit is continuous exactly when all coordinate composites are continuous Theorem
- Inverse limits of finite discrete groups are Hausdorff and totally disconnected, and compact assuming Choice Theorem
Dependency tree · two levels
15 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)