English
A commutative group with ultrametric norm is a nonarchimedean topological group: every neighborhood of 1 contains an open subgroup.
Русский
Коммутативная группа с ультраметрической нормой образует неабсолютно слабо аримедовую топологическую группу: каждая окрестность единицы содержит открытую подгруппу.
LaTeX
$$$\\text{NonarchimedeanGroup } M$ with the stated property$$
Lean4
/-- A commutative group with an ultrametric group seminorm is nonarchimedean (as a topological
group, i.e. every neighborhood of 1 contains an open subgroup). -/
@[to_additive /-- A commutative additive group with an ultrametric group seminorm is nonarchimedean
(as a topological group, i.e. every neighborhood of 0 contains an open subgroup). -/
]
instance nonarchimedeanGroup : NonarchimedeanGroup M where
is_nonarchimedean := by simpa only [Metric.mem_nhds_iff] using fun U ⟨ε, hεp, hεU⟩ ↦ ⟨ball_openSubgroup M hεp, hεU⟩
