ProCGroups.ProC.OpenNormalSubgroups.BasisAtOne
This module supplies the topological part of the construction. It checks continuity and stagewise neighborhood properties so that the completed object inherits the required topology.
import
- Mathlib.Topology.Algebra.ClopenNhdofOne
Imported by
- ProCGroups.Generation.Convergence
- ProCGroups.Generation.WordProductsAndClosure
- ProCGroups.LocalWeight.MetrizabilityAndQuotients
- ProCGroups.ProC.OpenNormalSubgroups
- ProCGroups.ProC.OpenNormalSubgroups.FilteredFamilies
- ProCGroups.ProC.OpenNormalSubgroups.ProCGroup
- ProCGroups.ProC.OpenNormalSubgroups.Separation
- ProCGroups.ProC.Quotients.ClosedSubgroupNeighborhoods
- ProCGroups.ProC.Quotients.LeftQuotientProjectionSections
theorem exists_openNormalSubgroup_sub_open_nhds_of_one [CompactSpace G]
[TotallyDisconnectedSpace G] {W : Set G} (hW : IsOpen W) (h1W : (1 : G) ∈ W) :
∃ U : OpenNormalSubgroup G, ((U : Subgroup G) : Set G) ⊆ WIn a compact totally disconnected topological group, any open neighborhood of \(1\) contains an open normal subgroup.
Show proof
by
simpa using ProfiniteGrp.exist_openNormalSubgroup_sub_open_nhds_of_one hW h1WProof. Work with open normal subgroups and finite quotients of the profinite or pro-\(C\) group. Finite quotients separate points, and continuous homomorphisms are determined by their compatible quotient-level maps. Closed-subgroup and subgroup-permanence claims use ambient open-normal approximation: an open normal subgroup of the closed subgroup is refined by the intersection with an ambient open normal subgroup of \(G\). Continuity is checked by composing with all finite quotient projections; each composite is a continuous map between finite or profinite quotient spaces. Compatibility under refinement then assembles the finite-quotient data and proves the required profinite or pro-\(C\) statement. The quotient-level assertions are stable under passing to smaller open normal subgroups, so they define compatible data in the inverse system of finite quotients. For pro-\(C\) permanence, the construction uses only operations for which the chosen finite group class is closed. Hence the finite verification assembles to the desired profinite map, subgroup, or quotient statement. For categorical constructions such as products, pullbacks, and inverse limits, the universal property is checked by composing with the coordinate projections. The uniqueness part follows because any two candidate maps with the same finite quotient composites have the same value on every separating finite quotient. Existence is obtained by assembling the compatible coordinate maps supplied by the finite quotient construction. The topological assertion is checked by the initial topology of the inverse limit. After composing with each finite-stage projection, the relevant map is a continuous finite-stage homomorphism or an operation on a finite product; compactness, Hausdorffness, total disconnectedness, and profiniteness are then inherited from the finite stages by the standard inverse-limit argument.
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