Theorems · Theorem · general topology
CompletelyRegularSpace.completely_regular
∀ {X : Type u} {inst : TopologicalSpace X} [self : CompletelyRegularSpace X] (x : X) (K : Set X),
IsClosed K → x ∉ K → ∃ f, Continuous f ∧ f x = 0 ∧ Set.EqOn f 1 K- Cited by
- 4 results in Mathlib
- Foundations
- Depth 114 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- CompletelyRegularSpace
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.
- Setstatement · cited by 53,352
- Realstatement · cited by 25,697
- TopologicalSpacestatement and proof · cited by 24,529
- Set.Elemstatement · cited by 7,166
- Continuousstatement · cited by 2,592
- IsClosedstatement · cited by 1,639
- unitIntervalstatement · cited by 607
- Set.EqOnstatement · cited by 603
- CompletelyRegularSpacestatement and proof · cited by 23
Cited by4
Results whose statement or proof uses this declaration.
- Topology.IsInducing.completelyRegularSpaceproof · cited by 2
- separatesPoints_continuous_of_t35Space_Iccproof · cited by 1
- CompletelyRegularSpace.exists_BCNNproof · cited by 1
- separatesPoints_continuous_of_t35Spaceproof · cited by 0