English
In a locally path-connected topological group, the path component of the identity is an open normal subgroup and is in fact clopen.
Русский
В локально путево связной топологической группе компонент пути равной единице является открытой нормальной подгруппой и на самом деле разрезана на открытое множество и замкнуто.
LaTeX
$$$\text{pathComponentOne } G \text{ is an open normal subgroup of } G$$$
Lean4
/-- The path component of the identity in a locally path connected topological group,
as an open normal subgroup. It is, in fact, clopen. -/
@[to_additive (attr := simps!) /--
The path component of the identity in a locally path connected additive topological group,
as an open normal additive subgroup. It is, in fact, clopen. -/
]
def pathComponentOne [Group G] [IsTopologicalGroup G] [LocPathConnectedSpace G] : OpenNormalSubgroup G
where
toSubgroup := .pathComponentOne G
isOpen' := .pathComponent 1
isNormal' := .pathComponentOne G
