The character of an open subgroup of index two is continuous #
A subgroup N of index two in a group G has a character χ_N : G →* Multiplicative (ZMod 2)
with kernel N, Subgroup.indexTwoCharacter. When G is a topological group and N is open,
that kernel is open, so χ_N is continuous; this is what makes χ_N a class in continuous
cohomology with 𝔽₂ coefficients.
Main results #
Subgroup.continuous_indexTwoCharacter: the character of an open subgroup of index two is continuous.
theorem
Subgroup.continuous_indexTwoCharacter
{G : Type u_1}
[Group G]
[TopologicalSpace G]
[SeparatelyContinuousMul G]
{N : Subgroup G}
(hN : N.index = 2)
(hNo : IsOpen ↑N)
:
Continuous ⇑(N.indexTwoCharacter hN)
The character of an open subgroup of index two is continuous: its kernel is the open
subgroup itself (Subgroup.ker_indexTwoCharacter).