Theorems · Theorem · general topology
IsClopen.preimage
∀ {X : Type u} {Y : Type v} [inst : TopologicalSpace X] [inst_1 : TopologicalSpace Y] {s : Set Y},
IsClopen s → ∀ {f : X → Y}, Continuous f → IsClopen (f ⁻¹' s)- Defined in
- Mathlib.Topology.Clopen
- Cited by
- 10 results in Mathlib
- Foundations
- Depth 59 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites7
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.preimagestatement · cited by 4,946
- Continuousstatement and proof · cited by 2,592
- IsClopenstatement and proof · cited by 189
- IsOpen.preimageproof · cited by 147
- IsClosed.preimageproof · cited by 138
Cited by10
Results whose statement or proof uses this declaration.
- preconnectedSpace_iff_clopenproof · cited by 3
- TopologicalSpace.Compacts.isClopen_singleton_botproof · cited by 3
- DiscreteQuotient.isClopen_preimageproof · cited by 3
- Profinite.exists_lift_of_finite_of_injective_of_surjectiveproof · cited by 1
- IsOpenMap.enatCard_connectedComponents_le_encard_preimage_singletonproof · cited by 1
- TopologicalSpace.Closeds.isClopen_singleton_botproof · cited by 1
- TopologicalSpace.Clopens.exists_prod_subsetproof · cited by 1
- IsOpenMap.finite_connectedComponents_of_finite_preimage_singletonproof · cited by 0