Theorems · Definition · general topology

ConnectedComponents.equivOfIsClopen

{α : Type u} →
  [inst : TopologicalSpace α] →
    {ι : Type u_3} →
      {U : ι → Set α} →
        (∀ (i : ι), IsClopen (U i)) →
          Pairwise (Function.onFun Disjoint U) →
            ⋃ i, U i = Set.univ → ConnectedComponents α ≃ (i : ι) × ConnectedComponents ↑(U i)

A pairwise disjoint cover by clopens partitions the connected components.

Defined in: Mathlib.Topology.Connected.Clopen
Cited by: 5 results in Mathlib
Foundations: Depth 84 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: TopologicalSpace

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Cites14

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
Equivstatement · cited by 8,337
Set.Elemstatement and proof · cited by 7,166
Set.univstatement and proof · cited by 3,945
Equiv.symmproof · cited by 3,681
Set.iUnionstatement and proof · cited by 2,483
Disjointstatement and proof · cited by 2,201
Function.onFunstatement and proof · cited by 570
Pairwisestatement and proof · cited by 516
IsClopenstatement and proof · cited by 189
Equiv.ofBijectiveproof · cited by 70

Cited by6

Results whose statement or proof uses this declaration.

ConnectedComponents.equivOfIsClopenOfIsConnectedproof · cited by 1
ConnectedComponents.equivOfIsClopen_mkstatement and proof · cited by 1
ConnectedComponents.exists_fun_isClopen_of_infiniteproof · cited by 1
ConnectedComponents.equivOfIsClopen_symm_mkstatement · cited by 0
IsOpenMap.finite_connectedComponents_of_finite_preimage_singletonproof · cited by 0
ConnectedComponents.equivOfIsClopen.congr_simpstatement and proof · cited by 0