English
If G is a topological group and U is an open subgroup, then the quotient G/U carries the discrete topology.
Русский
Если G — топологическая группа и U — открытая подгруппа, то фактор-город G/U имеет дискретную топологию.
LaTeX
$$$$ \\text{DiscreteTopology}(G/U). $$$$
Lean4
@[to_additive]
theorem discreteTopology [ContinuousMul G] (U : Subgroup G) (h : IsOpen (U : Set G)) : DiscreteTopology (G ⧸ U) :=
by
refine discreteTopology_iff_isOpen_singleton.mpr (fun g ↦ ?_)
induction g using Quotient.inductionOn with
| h g =>
change IsOpen (QuotientGroup.mk ⁻¹' {QuotientGroup.mk g})
convert_to IsOpen ((g * ·) '' U)
· ext g'
simp only [Set.mem_preimage, Set.mem_singleton_iff, QuotientGroup.eq, Set.image_mul_left]
rw [← U.inv_mem_iff]
simp
· exact Homeomorph.mulLeft g |>.isOpen_image |>.mpr h
