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Theorems · Theorem · general topology

isClopen_connectedComponent

∀ {α : Type u} [inst : TopologicalSpace α] [LocallyConnectedSpace α] {x : α}, IsClopen (connectedComponent x)
Defined in
Mathlib.Topology.Connected.LocallyConnected
Cited by
1 results in Mathlib
Foundations
Depth 83 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes
TopologicalSpaceLocallyConnectedSpace
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Around this declaration

Dashed lines are statement dependencies; solid lines are citations in proofs.

IsLocallyConstant.of_constant_on_connected_clopens · cited by 1IsLocallyConstant.of_cons…TopologicalSpace · cited by 24529TopologicalSpaceIsClopen · cited by 189IsClopenconnectedComponent · cited by 68connectedComponentLocallyConnectedSpace · cited by 26LocallyConnectedSpaceisClosed_connectedComponent · cited by 3isClosed_connectedCompone…isOpen_connectedComponent · cited by 3isOpen_connectedComponentisClopen_connectedComponentCITED BYCITES

Cites6

Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

Cited by1

Results whose statement or proof uses this declaration.