Theorems · Theorem · general topology
IsDiscrete.iUnion
∀ {X : Type u_1} [inst : TopologicalSpace X] {ι : Sort u_3} [Finite ι] {s : ι → Set X},
(∀ (i : ι), IsDiscrete (s i)) → (∀ (i : ι), IsClosed (s i)) → IsDiscrete (⋃ i, s i)The union of finitely many discrete closed subsets is discrete.
- Defined in
- Mathlib.Topology.DiscreteSubset
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 82 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- TopologicalSpaceFinite
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
- Finitestatement and proof · cited by 3,029
- Compl.complproof · cited by 2,925
- Set.iUnionstatement and proof · cited by 2,483
- IsClosedstatement and proof · cited by 1,639
- IsDiscretestatement and proof · cited by 86
- Filter.codiscreteproof · cited by 34
- Set.compl_iUnionproof · cited by 32
- compl_mem_codiscrete_iffproof · cited by 2
Cited by3
Results whose statement or proof uses this declaration.
- IsDiscrete.biUnionproof · cited by 1
- IsDiscrete.unionproof · cited by 1
- discreteTopology_iUnion_finiteproof · cited by 0