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Theorems · Theorem · general topology

Filter.HasBasis.eq_bot_iff

∀ {α : Type u_1} {ι : Sort u_4} {l : Filter α} {p : ι → Prop} {s : ι → Set α},
  l.HasBasis p s → (l = ⊥ ↔ ∃ i, p i ∧ s i = ∅)
Defined in
Mathlib.Order.Filter.Bases.Basic
Cited by
4 results in Mathlib
Foundations
Depth 55 from the axioms · uses propext, Classical.choice, Quot.sound
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Around this declaration

Dashed lines are statement dependencies; solid lines are citations in proofs.

Filter.cocompact_eq_bot · cited by 5Filter.cocompact_eq_botFilter.HasBasis.liminf_eq_sSup_univ_of_empty · cited by 2HasBasis.liminf_eq_sSup_u…nhdsGT_eq_bot_iff · cited by 1nhdsGT_eq_bot_iffFilter.coLindelof_eq_bot · cited by 0Filter.coLindelof_eq_botSet · cited by 53352SetFilter · cited by 8121FilterBot.bot · cited by 4720Bot.botFilter.HasBasis · cited by 604Filter.HasBasisnot_iff_not · cited by 159not_iff_notFilter.neBot_iff · cited by 27Filter.neBot_iffFilter.HasBasis.neBot_iff · cited by 5HasBasis.neBot_iffHasBasis.eq_bot_iffCITED BYCITES

Cites7

Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

Cited by4

Results whose statement or proof uses this declaration.