English
The valuation-based topology is compatible with ring homomorphisms preserving valuation structure.
Русский
Топология, заданная оценкой, совместима с гомоморфизмами колец, сохраняющими структуру оценки.
LaTeX
$$$\varphi:R\to R' \text{ ring homomorphism with } v'\circ\varphi = v$$$
Lean4
theorem continuous_valuation : Continuous v :=
by
simp only [continuous_iff_continuousAt, ContinuousAt]
rintro x
by_cases hx : v x = 0
·
simpa [hx, (hasBasis_nhds _).tendsto_iff WithZeroTopology.hasBasis_nhds_zero,
Valuation.map_sub_of_right_eq_zero _ hx] using fun i hi ↦ ⟨.mk0 i hi, fun y ↦ id⟩
·
simpa [(hasBasis_nhds _).tendsto_iff (WithZeroTopology.hasBasis_nhds_of_ne_zero hx)] using
⟨.mk0 (v x) hx, fun _ ↦ Valuation.map_eq_of_sub_lt _⟩