English
A space α is T0 if and only if the kernel of the uniformity 𝓤α equals the diagonal, i.e., (𝓤α).ker = diagonal α.
Русский
Пространство α является T0 тогда, когда ядро равномерности равно диагонали: (𝓤α).ker = диагональ(α).
LaTeX
$$$T0Space(α) \iff (𝓤(α)).ker = \Delta(α)$$$
Lean4
theorem t0Space_iff_ker_uniformity : T0Space α ↔ (𝓤 α).ker = diagonal α :=
by
simp_rw [t0Space_iff_uniformity, subset_antisymm_iff, diagonal_subset_iff, subset_def, Prod.forall, Filter.mem_ker,
mem_diagonal_iff, iff_self_and]
exact fun _ x s hs ↦ refl_mem_uniformity hs