Theorems · Definition · general topology

Topology.IsCoinducing.connectedComponentsHomeomorph

{X : Type u_1} →
  {Y : Type u_2} →
    [inst : TopologicalSpace X] →
      [inst_1 : TopologicalSpace Y] →
        {f : X → Y} →
          Topology.IsCoinducing f →
            (∀ (y : Y), IsConnected (f ⁻¹' {y})) → ConnectedComponents X ≃ₜ ConnectedComponents Y

If f : X → Y is coinducing and has connected fibers, it induces a homeomorphism on π₀.

Defined in: Mathlib.Topology.Homeomorph.Lemmas
Cited by: 3 results in Mathlib
Foundations: Depth 88 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: TopologicalSpace TopologicalSpace

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Cites10

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
Homeomorphstatement · cited by 725
IsConnectedstatement and proof · cited by 116
ConnectedComponentsstatement · cited by 40
Topology.IsCoinducingstatement and proof · cited by 31
IsHomeomorph.homeomorphproof · cited by 18
Continuous.connectedComponentsMapproof · cited by 7
Topology.IsCoinducing.continuousproof · cited by 6

Cited by4

Results whose statement or proof uses this declaration.

AlgebraicGeometry.Scheme.Hom.connectedComponentsHomeomorphproof · cited by 1
Topology.IsCoinducing.connectedComponentsHomeomorph.congr_simpstatement and proof · cited by 0
Topology.IsCoinducing.connectedComponentsHomeomorph_mkstatement · cited by 0
Topology.IsCoinducing.connectedComponentsHomeomorph_symm_mk_applystatement and proof · cited by 0