jointlySurjectiveTopology — Mathlib

English

Alternate equivalent form of mem_grothendieckTopology_iff stating the equivalence via a simpler characterization using the possessive of R and S.

Русский

Альтернативная эквивалентная форма mem_grothendieckTopology_iff через более простой признак R и S.

LaTeX

$$Same as above with simplifications$$

Lean4

/-- The jointly surjective topology on `Scheme` is defined by the same condition as the jointly
surjective pretopology. -/
def jointlySurjectiveTopology : GrothendieckTopology Scheme.{u} :=
  jointlySurjectivePretopology.toGrothendieck.copy (fun X s ↦ jointlySurjectivePretopology X ↑s) <|
    funext fun _ ↦
      Set.ext fun s ↦
        ⟨fun ⟨_, hp, hps⟩ x ↦
          let ⟨Y, u, hu, hmem⟩ := hp x;
          ⟨Y, u, Presieve.map_monotone hps _ hu, hmem⟩,
          fun hs ↦ ⟨s, hs, le_rfl⟩⟩

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