Theorems · Theorem · general topology

ContinuousMap.dist_le

∀ {α : Type u_1} {β : Type u_2} [inst : TopologicalSpace α] [inst_1 : CompactSpace α] [inst_2 : PseudoMetricSpace β]
  {f g : C(α, β)} {C : ℝ}, 0 ≤ C → (dist f g ≤ C ↔ ∀ (x : α), dist (f x) (g x) ≤ C)

The distance between two functions is controlled by the supremum of the pointwise distances.

Defined in

Mathlib.Topology.ContinuousMap.Compact

Cited by

3 results in Mathlib

Foundations

Depth 126 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

TopologicalSpaceCompactSpacePseudoMetricSpace