Theorems · Theorem · general topology

IsPreconnected.intermediate_value_Iii

∀ {X : Type u} {α : Type v} [inst : TopologicalSpace X] [inst_1 : LinearOrder α] [inst_2 : TopologicalSpace α]
  [OrderClosedTopology α] {s : Set X},
  IsPreconnected s →
    ∀ {l₁ l₂ : Filter X} [l₁.NeBot] [l₂.NeBot],
      l₁ ≤ Filter.principal s →
        l₂ ≤ Filter.principal s →
          ∀ {f : X → α},
            ContinuousOn f s → Filter.Tendsto f l₁ Filter.atBot → Filter.Tendsto f l₂ Filter.atTop → Set.univ ⊆ f '' s

Defined in

Mathlib.Topology.Order.IntermediateValue

Cited by

0 results in Mathlib

Foundations

Depth 81 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

TopologicalSpaceLinearOrderTopologicalSpaceOrderClosedTopologyFilter.NeBotFilter.NeBot