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Theorems · Theorem · general topology

limsSup_eq_of_le_nhds

∀ {α : Type u_2} [inst : ConditionallyCompleteLinearOrder α] [inst_1 : TopologicalSpace α] [OrderTopology α]
  {f : Filter α} {a : α} [f.NeBot], f ≤ nhds a → f.limsSup = a

If a filter is converging, its liminf coincides with its limit.

Defined in
Mathlib.Topology.Order.LiminfLimsup
Cited by
2 results in Mathlib
Foundations
Depth 78 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes
ConditionallyCompleteLinearOrderTopologicalSpaceOrderTopologyFilter.NeBot
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Around this declaration

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

Filter.Tendsto.limsup_eq · cited by 12Tendsto.limsup_eqClusterPt.le_limsSup · cited by 3ClusterPt.le_limsSupTopologicalSpace · cited by 24529TopologicalSpaceFilter · cited by 8121Filternhds · cited by 5554nhdsOrderTopology · cited by 1355OrderTopologyFilter.NeBot · cited by 853Filter.NeBotConditionallyCompleteLinearOrder · cited by 542ConditionallyCompleteLine…Filter.limsSup · cited by 29Filter.limsSuplimsInf_eq_of_le_nhds · cited by 2limsInf_eq_of_le_nhdslimsSup_eq_of_le_nhdsCITED BYCITES

Cites8

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Cited by2

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