Theorems · Theorem · general topology

IsPreconnected.intermediate_value_Ioc

∀ {X : Type u} {α : Type v} [inst : TopologicalSpace X] [inst_1 : LinearOrder α] [inst_2 : TopologicalSpace α]
  [OrderClosedTopology α] {s : Set X},
  IsPreconnected s →
    ∀ {a : X} {l : Filter X},
      a ∈ s →
        ∀ [l.NeBot],
          l ≤ Filter.principal s →
            ∀ {f : X → α}, ContinuousOn f s → ∀ {v : α}, Filter.Tendsto f l (nhds v) → Set.Ioc v (f a) ⊆ f '' s

Defined in

Mathlib.Topology.Order.IntermediateValue

Cited by

2 results in Mathlib

Foundations

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

Assumes

TopologicalSpaceLinearOrderTopologicalSpaceOrderClosedTopologyFilter.NeBot