Theorems · Theorem · general topology
BoundedContinuousFunction.dist_le
∀ {α : Type u} {β : Type v} [inst : TopologicalSpace α] [inst_1 : PseudoMetricSpace β]
{f g : BoundedContinuousFunction 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.
- Cited by
- 11 results in Mathlib
- Foundations
- Depth 118 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites9
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement and proof · cited by 62,936
- Realstatement and proof · cited by 25,697
- TopologicalSpacestatement and proof · cited by 24,529
- PseudoMetricSpacestatement and proof · cited by 1,550
- Dist.diststatement and proof · cited by 1,539
- le_transproof · cited by 985
- BoundedContinuousFunctionstatement and proof · cited by 511
- csInf_leproof · cited by 51
- BoundedContinuousFunction.dist_coe_le_distproof · cited by 18
Cited by11
Results whose statement or proof uses this declaration.
- BoundedContinuousFunction.norm_leproof · cited by 11
- BoundedContinuousFunction.dist_le_iff_of_nonemptyproof · cited by 3
- ContinuousMap.dist_leproof · cited by 3
- BoundedContinuousFunction.tendsto_iff_tendstoUniformlyproof · cited by 2
- BoundedContinuousFunction.lipschitz_compproof · cited by 2
- BoundedContinuousFunction.lipschitz_compContinuousproof · cited by 2
- BoundedContinuousFunction.dist_extend_extendproof · cited by 1
- TopologicalSpace.exists_isInducing_l_inftyproof · cited by 1
- ContinuousMap.isUniformInducing_equivBoundedOfCompactproof · cited by 1
- BoundedContinuousFunction.tietze_extension_stepproof · cited by 1
- BoundedContinuousFunction.arzela_ascoli₁proof · cited by 1