Theorems · Theorem · general topology

Continuous.exists_forall_le_of_hasCompactSupport

∀ {α : Type u_2} {β : Type u_3} [inst : LinearOrder α] [inst_1 : TopologicalSpace α] [inst_2 : TopologicalSpace β]
  [ClosedIicTopology α] [Nonempty β] [inst_5 : Zero α] {f : β → α},
  Continuous f → HasCompactSupport f → ∃ x, ∀ (y : β), f x ≤ f y

A continuous function with compact support has a global minimum.

Defined in

Mathlib.Topology.Order.Compact

Cited by

1 results in Mathlib

Foundations

Depth 88 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

LinearOrderTopologicalSpaceTopologicalSpaceClosedIicTopologyNonemptyZero