Theorems · Theorem · general topology
IsLocallyConstant.apply_eq_of_isPreconnected
∀ {X : Type u_1} {Y : Type u_2} [inst : TopologicalSpace X] {f : X → Y},
IsLocallyConstant f → ∀ {s : Set X}, IsPreconnected s → ∀ {x y : X}, x ∈ s → y ∈ s → f x = f yA locally constant function is constant on any preconnected set.
- Defined in
- Mathlib.Topology.LocallyConstant.Basic
- Cited by
- 5 results in Mathlib
- Foundations
- Depth 58 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- TopologicalSpace
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites10
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement and proof · cited by 53,352
- TopologicalSpacestatement and proof · cited by 24,529
- Set.preimageproof · cited by 4,946
- Compl.complproof · cited by 2,925
- Set.Nonemptyproof · cited by 2,627
- IsPreconnectedstatement and proof · cited by 205
- Set.union_compl_selfproof · cited by 57
- IsLocallyConstantstatement and proof · cited by 51
- Set.inter_emptyproof · cited by 36
- Set.inter_compl_selfproof · cited by 17
Cited by5
Results whose statement or proof uses this declaration.
- IsLocallyConstant.apply_eq_of_preconnectedSpaceproof · cited by 2
- LocallyConstant.apply_eq_of_preconnectedSpaceproof · cited by 1
- eq_of_germ_isConstantproof · cited by 1
- LocallyConstant.apply_eq_of_isPreconnectedproof · cited by 0
- IsLocallyConstant.iff_is_constproof · cited by 0