Theorems · Theorem · general topology

IsCompact.exists_isMinOn

• 1000+ list: Extreme value theorem

∀ {α : Type u_2} {β : Type u_3} [inst : LinearOrder α] [inst_1 : TopologicalSpace α] [inst_2 : TopologicalSpace β]
  [ClosedIicTopology α] {s : Set β}, IsCompact s → s.Nonempty → ∀ {f : β → α}, ContinuousOn f s → ∃ x ∈ s, IsMinOn f s x

The extreme value theorem: a continuous function realizes its minimum on a compact set.

Defined in

Mathlib.Topology.Order.Compact

Cited by

15 results in Mathlib

Foundations

Depth 79 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

LinearOrderTopologicalSpaceTopologicalSpaceClosedIicTopology