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Theorems · Theorem · general topology

LocallyConnectedSpace.open_connected_basis

∀ {α : Type u_3} {inst : TopologicalSpace α} [self : LocallyConnectedSpace α] (x : α),
  (nhds x).HasBasis (fun s => IsOpen s ∧ x ∈ s ∧ IsConnected s) id

Open connected neighborhoods form a basis of the neighborhoods filter.

Defined in
Mathlib.Topology.Connected.LocallyConnected
Cited by
6 results in Mathlib
Foundations
Depth 19 from the axioms · uses propext, Quot.sound
Assumes
LocallyConnectedSpace
Mathlib docsSourceLoogleLeanSearch

Around this declaration

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

Topology.IsOpenEmbedding.locallyConnectedSpace · cited by 2IsOpenEmbedding.locallyCo…connectedComponentIn_mem_nhds · cited by 2connectedComponentIn_mem_…TopologicalSpace.IsTopologicalBasis.isOpen_isPreconnected · cited by 2IsTopologicalBasis.isOpen…ChartedSpace.locallyConnectedSpace · cited by 0ChartedSpace.locallyConne…locallyConnectedSpace_iff_hasBasis_isOpen_isConnected · cited by 0locallyConnectedSpace_iff…Homeomorph.locallyConnectedSpace · cited by 0Homeomorph.locallyConnect…Set · cited by 53352SetTopologicalSpace · cited by 24529TopologicalSpacenhds · cited by 5554nhdsIsOpen · cited by 2400IsOpenFilter.HasBasis · cited by 604Filter.HasBasisIsConnected · cited by 116IsConnectedLocallyConnectedSpace · cited by 26LocallyConnectedSpaceLocallyConnectedSpace.open_co…CITED BYCITES

Cites7

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Cited by6

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