ProCGroups.Profinite.Basic
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.
Imported by
- ProCGroups
- ProCGroups.Categorical.ProfinitePullbacks
- ProCGroups.Categorical.PushoutSquares
- ProCGroups.Duality
- ProCGroups.Generation.Convergence
- ProCGroups.Generation.QuotientCriteria
- ProCGroups.Generation.WordProductsAndClosure
- ProCGroups.InverseSystems.ProfiniteLimits
- ProCGroups.Order.Basic
- ProCGroups.ProC
- ProCGroups.ProC.OpenNormalSubgroups.CountableChains
- ProCGroups.ProC.OpenNormalSubgroups.ProCGroup
- ProCGroups.ProC.OpenNormalSubgroups.Separation
- ProCGroups.Profinite
- ProCGroups.Profinite.MathlibBridge
def IsProfiniteGroup (G : Type u) [Group G] [TopologicalSpace G] : Prop :=
IsTopologicalGroup G ∧ CompactSpace G ∧ T2Space G ∧ TotallyDisconnectedSpace GAn unbundled profinite group is a compact Hausdorff totally disconnected topological group.
theorem isTopologicalGroup (hG : IsProfiniteGroup G) : IsTopologicalGroup GA profinite group is a topological group.
Show proof
hG.1Proof. 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. For pro-\(C\) claims, the resulting finite quotients lie in \(C\) by the closure property used in the construction: closure under quotients, subgroups, finite products, or extensions. 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.
□theorem compactSpace (hG : IsProfiniteGroup G) : CompactSpace GThe compact-space instance on a profinite group.
Show proof
hG.2.1Proof. 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. For pro-\(C\) claims, the resulting finite quotients lie in \(C\) by the closure property used in the construction: closure under quotients, subgroups, finite products, or extensions. 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. Functoriality and naturality are proved by comparing the two composites after projection to an arbitrary target stage. The source stage is the pullback or refined quotient required by the construction, and both composites are induced by the same quotient homomorphism together with the same coefficient map; hence the coordinate maps agree and so do the completed maps.
□theorem t2Space (hG : IsProfiniteGroup G) : T2Space GThe Hausdorff space structure on a profinite group.
Show proof
hG.2.2.1Proof. 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. For pro-\(C\) claims, the resulting finite quotients lie in \(C\) by the closure property used in the construction: closure under quotients, subgroups, finite products, or extensions. 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.
□theorem t1Space (hG : IsProfiniteGroup G) : T1Space GA profinite group is \(T_1\).
Show proof
by
letI : T2Space G := hG.t2Space
infer_instanceProof. 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. For pro-\(C\) claims, the resulting finite quotients lie in \(C\) by the closure property used in the construction: closure under quotients, subgroups, finite products, or extensions. 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. Functoriality and naturality are proved by comparing the two composites after projection to an arbitrary target stage. The source stage is the pullback or refined quotient required by the construction, and both composites are induced by the same quotient homomorphism together with the same coefficient map; hence the coordinate maps agree and so do the completed maps.
□theorem totallyDisconnectedSpace (hG : IsProfiniteGroup G) : TotallyDisconnectedSpace GA profinite group is totally disconnected.
Show proof
hG.2.2.2Proof. 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. For pro-\(C\) claims, the resulting finite quotients lie in \(C\) by the closure property used in the construction: closure under quotients, subgroups, finite products, or extensions. 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.
□theorem of_isProfiniteSpace [IsTopologicalGroup G]
(hG : InverseSystems.IsProfiniteSpace G) : IsProfiniteGroup GA topological group whose underlying topological space is profinite is a profinite group.
Show proof
⟨inferInstance, hG.1, hG.2.1, hG.2.2⟩Proof. 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. For pro-\(C\) claims, the resulting finite quotients lie in \(C\) by the closure property used in the construction: closure under quotients, subgroups, finite products, or extensions. 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.
□theorem of_isClosed_subgroup_of_closedSubgroup
{P : Subgroup G → Prop}
(h : ∀ H : ClosedSubgroup G, P (H : Subgroup G))
(H : Subgroup G) (hH : IsClosed (H : Set G)) :
P HRepackage a permanence theorem formulated for closed subgroups into the ordinary subgroup form.
Show proof
by
let HC : ClosedSubgroup G := ⟨H, hH⟩
simpa using h HCProof. 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 \(G\). 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. For quotient and subgroup claims, representatives are chosen in the ambient group and the induced map is checked to send the class of an element to the class of its image. Normality, openness, and membership in the finite quotient class are preserved by the subgroup, quotient, intersection, or inverse-image closure property being invoked.
□theorem of_finite_discrete (G : Type u) [Group G] [TopologicalSpace G]
[Finite G] [DiscreteTopology G] : IsProfiniteGroup GAny finite discrete topological group is profinite.
Show proof
by
letI : Fintype G := Fintype.ofFinite G
letI : CompactSpace G := by infer_instance
letI : T2Space G := by infer_instance
letI : TotallyDisconnectedSpace G := by infer_instance
exact ⟨inferInstance, inferInstance, inferInstance, inferInstance⟩Proof. 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. For pro-\(C\) claims, the resulting finite quotients lie in \(C\) by the closure property used in the construction: closure under quotients, subgroups, finite products, or extensions. 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.
□theorem quotient_openNormalSubgroup (hG : IsProfiniteGroup G) (U : OpenNormalSubgroup G) :
IsProfiniteGroup (G ⧸ (U : Subgroup G))A quotient of a profinite group by an open normal subgroup is profinite.
Show proof
by
letI : CompactSpace G := hG.compactSpace
letI : T2Space G := hG.t2Space
letI : Finite (G ⧸ (U : Subgroup G)) := openNormalSubgroup_finiteQuotient (G := G) U
letI : DiscreteTopology (G ⧸ (U : Subgroup G)) :=
QuotientGroup.discreteTopology (openNormalSubgroup_isOpen (G := G) U)
exact of_finite_discrete (G := G ⧸ (U : Subgroup G))Proof. 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 \(G\). For pro-\(C\) claims, the resulting finite quotients lie in \(C\) by the closure property used in the construction: closure under quotients, subgroups, finite products, or extensions. 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.
□theorem of_closedSubgroup (hG : IsProfiniteGroup G) (H : ClosedSubgroup G) :
IsProfiniteGroup ↥(H : Subgroup G)A closed subgroup of a profinite group is profinite.
Show proof
by
letI : CompactSpace G := IsProfiniteGroup.compactSpace hG
letI : T2Space G := IsProfiniteGroup.t2Space hG
letI : TotallyDisconnectedSpace G := IsProfiniteGroup.totallyDisconnectedSpace hG
letI : CompactSpace H := inferInstance
exact ⟨inferInstance, inferInstance, inferInstance, inferInstance⟩Proof. 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 \(G\). For pro-\(C\) claims, the resulting finite quotients lie in \(C\) by the closure property used in the construction: closure under quotients, subgroups, finite products, or extensions. 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. 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. For quotient and subgroup claims, representatives are chosen in the ambient group and the induced map is checked to send the class of an element to the class of its image. Normality, openness, and membership in the finite quotient class are preserved by the subgroup, quotient, intersection, or inverse-image closure property being invoked.
□theorem of_isClosed_subgroup (hG : IsProfiniteGroup G) (H : Subgroup G)
(hH : IsClosed (H : Set G)) : IsProfiniteGroup ↥HA closed ordinary subgroup of a profinite group is profinite with the induced topology.
Show proof
by
exact of_isClosed_subgroup_of_closedSubgroup
(G := G) (P := fun H => IsProfiniteGroup ↥H) (of_closedSubgroup (G := G) hG) H hHProof. 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 \(G\). For pro-\(C\) claims, the resulting finite quotients lie in \(C\) by the closure property used in the construction: closure under quotients, subgroups, finite products, or extensions. 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.
□theorem pi {α : Type v} {β : α → Type u}
[∀ a, Group (β a)] [∀ a, TopologicalSpace (β a)] [∀ a, IsTopologicalGroup (β a)]
(hβ : ∀ a, IsProfiniteGroup (β a)) :
IsProfiniteGroup ((a : α) → β a)Arbitrary product permanence for profinite groups.
Show proof
by
letI : ∀ a, CompactSpace (β a) := fun a => IsProfiniteGroup.compactSpace (hβ a)
letI : ∀ a, T2Space (β a) := fun a => IsProfiniteGroup.t2Space (hβ a)
letI : ∀ a, TotallyDisconnectedSpace (β a) := fun a =>
IsProfiniteGroup.totallyDisconnectedSpace (hβ a)
exact ⟨inferInstance, inferInstance, inferInstance, inferInstance⟩Proof. 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. For pro-\(C\) claims, the resulting finite quotients lie in \(C\) by the closure property used in the construction: closure under quotients, subgroups, finite products, or extensions. 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.
□theorem prod {G : Type u} [Group G] [TopologicalSpace G] [IsTopologicalGroup G] {H : Type v}
[Group H] [TopologicalSpace H] [IsTopologicalGroup H]
(hG : IsProfiniteGroup G) (hH : IsProfiniteGroup H) :
IsProfiniteGroup (G × H)Binary-product case of IsProfiniteGroup.pi.
Show proof
by
letI : CompactSpace G := IsProfiniteGroup.compactSpace hG
letI : CompactSpace H := IsProfiniteGroup.compactSpace hH
letI : T2Space G := IsProfiniteGroup.t2Space hG
letI : T2Space H := IsProfiniteGroup.t2Space hH
letI : TotallyDisconnectedSpace G := IsProfiniteGroup.totallyDisconnectedSpace hG
letI : TotallyDisconnectedSpace H := IsProfiniteGroup.totallyDisconnectedSpace hH
exact ⟨inferInstance, inferInstance, inferInstance, inferInstance⟩Proof. 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. For pro-\(C\) claims, the resulting finite quotients lie in \(C\) by the closure property used in the construction: closure under quotients, subgroups, finite products, or extensions. 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.
□theorem eq_one_of_mem_all_openNormalSubgroups [CompactSpace G]
[TotallyDisconnectedSpace G] {x : G}
(hx : ∀ U : OpenNormalSubgroup G, x ∈ (U : Subgroup G)) : x = 1In a profinite group, an element lying in every open normal subgroup must be \(1\).
Show proof
by
by_contra hxne
let W : Set G := ({x} : Set G)ᶜ
have hW : IsOpen W := by
simp only [isOpen_compl_iff, finite_singleton, Finite.isClosed, W]
have h1W : (1 : G) ∈ W := by
have hx1 : (1 : G) ≠ x := by
intro h1x
exact hxne h1x.symm
simp only [mem_compl_iff, mem_singleton_iff, hx1, not_false_eq_true, W]
rcases ProfiniteGrp.exist_openNormalSubgroup_sub_open_nhds_of_one
(G := G) hW h1W with ⟨U, hUW⟩
have hxU : x ∈ (U : Subgroup G) := hx U
have hxW : x ∈ W := hUW hxU
simp only [mem_compl_iff, mem_singleton_iff, not_true_eq_false, W] at hxWProof. 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 \(G\). For pro-\(C\) claims, the resulting finite quotients lie in \(C\) by the closure property used in the construction: closure under quotients, subgroups, finite products, or extensions. Kernel and image statements are verified after quotienting by sufficiently small open normal subgroups, where they become ordinary finite group calculations. 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. 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. For quotient and subgroup claims, representatives are chosen in the ambient group and the induced map is checked to send the class of an element to the class of its image. Normality, openness, and membership in the finite quotient class are preserved by the subgroup, quotient, intersection, or inverse-image closure property being invoked.
□