Theorems · Theorem · general topology
IsOpenMap.finite_connectedComponents_of_finite_preimage_singleton_of_connectedSpace
∀ {X : Type u_1} {Y : Type u_2} [inst : TopologicalSpace X] [inst_1 : TopologicalSpace Y] {f : X → Y},
IsOpenMap f → IsClosedMap f → ∀ [ConnectedSpace Y] {y : Y}, (f ⁻¹' {y}).Finite → Finite (ConnectedComponents X)- Cited by
- 1 results in Mathlib
- Foundations
- Depth 102 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites13
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement · cited by 53,352
- TopologicalSpacestatement and proof · cited by 24,529
- Set.preimagestatement and proof · cited by 4,946
- Finitestatement · cited by 3,029
- Set.Finitestatement and proof · cited by 1,814
- lt_of_le_of_ltproof · cited by 432
- IsOpenMapstatement and proof · cited by 253
- IsClosedMapstatement and proof · cited by 138
- ConnectedComponentsstatement · cited by 40
- ConnectedSpacestatement and proof · cited by 37
- Set.Finite.encard_lt_topproof · cited by 10
- ENat.card_lt_topproof · cited by 4
Cited by1
Results whose statement or proof uses this declaration.
- IsOpenMap.finite_connectedComponents_of_finite_preimage_singletonproof · cited by 0