Theorems · Theorem · general topology
DiscreteQuotient.isClopen_preimage
∀ {X : Type u_2} [inst : TopologicalSpace X] (S : DiscreteQuotient X) (A : Set (Quotient S.toSetoid)),
IsClopen (S.proj ⁻¹' A)- Defined in
- Mathlib.Topology.DiscreteQuotient
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 72 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.preimagestatement · cited by 4,946
- IsClopenstatement · cited by 189
- DiscreteQuotientstatement and proof · cited by 65
- DiscreteQuotient.toSetoidstatement and proof · cited by 45
- DiscreteQuotient.projstatement · cited by 23
- IsClopen.preimageproof · cited by 10
- isClopen_discreteproof · cited by 8
- DiscreteQuotient.proj_continuousproof · cited by 2
Cited by3
Results whose statement or proof uses this declaration.
- DiscreteQuotient.isClopen_setOfPred_relproof · cited by 1
- DiscreteQuotient.isClosed_preimageproof · cited by 1
- DiscreteQuotient.isOpen_preimageproof · cited by 0