File: Mathlib/Topology/Connected/LocallyConnected.lean

full_name: LocallyConnectedSpace
code: /-- A topological space is **locally connected** if each neighborhood filter admits a basis
of connected *open* sets. Note that it is equivalent to each point having a basis of connected
(non necessarily open) sets but in a non-trivial way, so we choose this definition and prove the
equivalence later in `locallyConnectedSpace_iff_connected_basis`. -/
class LocallyConnectedSpace (α : Type*) [TopologicalSpace α] : Prop where
  /-- Open connected neighborhoods form a basis of the neighborhoods filter. -/
  open_connected_basis : ∀ x, (𝓝 x).HasBasis (fun s : Set α => IsOpen s ∧ x ∈ s ∧ IsConnected s) id
goal: Prop

full_name: LocallyConnectedSpace.open_connected_basis
code: /-- A topological space is **locally connected** if each neighborhood filter admits a basis
of connected *open* sets. Note that it is equivalent to each point having a basis of connected
(non necessarily open) sets but in a non-trivial way, so we choose this definition and prove the
equivalence later in `locallyConnectedSpace_iff_connected_basis`. -/
class LocallyConnectedSpace (α : Type*) [TopologicalSpace α] : Prop where
  /-- Open connected neighborhoods form a basis of the neighborhoods filter. -/
  open_connected_basis : ∀ x, (𝓝 x).HasBasis (fun s : Set α => IsOpen s ∧ x ∈ s ∧ IsConnected s) id
goal: (𝓝 x).HasBasis (fun s => IsOpen s ∧ x ∈ s ∧ IsConnected s) id

full_name: locallyConnectedSpace_iff_open_connected_basis
code: theorem locallyConnectedSpace_iff_open_connected_basis :
    LocallyConnectedSpace α ↔
      ∀ x, (𝓝 x).HasBasis (fun s : Set α => IsOpen s ∧ x ∈ s ∧ IsConnected s) id 
goal: LocallyConnectedSpace α ↔ ∀ (x : α), (𝓝 x).HasBasis (fun s => IsOpen s ∧ x ∈ s ∧ IsConnected s) id

full_name: locallyConnectedSpace_iff_open_connected_subsets
code: theorem locallyConnectedSpace_iff_open_connected_subsets :
    LocallyConnectedSpace α ↔
      ∀ x, ∀ U ∈ 𝓝 x, ∃ V : Set α, V ⊆ U ∧ IsOpen V ∧ x ∈ V ∧ IsConnected V 
goal: LocallyConnectedSpace α ↔ ∀ (x : α), ∀ U ∈ 𝓝 x, ∃ V ⊆ U, IsOpen V ∧ x ∈ V ∧ IsConnected V

full_name: DiscreteTopology.toLocallyConnectedSpace
code: /-- A space with discrete topology is a locally connected space. -/
instance (priority := 100) DiscreteTopology.toLocallyConnectedSpace (α) [TopologicalSpace α]
    [DiscreteTopology α] : LocallyConnectedSpace α 
goal: LocallyConnectedSpace α

full_name: connectedComponentIn_mem_nhds
code: theorem connectedComponentIn_mem_nhds [LocallyConnectedSpace α] {F : Set α} {x : α} (h : F ∈ 𝓝 x) :
    connectedComponentIn F x ∈ 𝓝 x 
goal: connectedComponentIn F x ∈ 𝓝 x

full_name: IsOpen.connectedComponentIn
code: protected theorem IsOpen.connectedComponentIn [LocallyConnectedSpace α] {F : Set α} {x : α}
    (hF : IsOpen F) : IsOpen (connectedComponentIn F x) 
goal: IsOpen (connectedComponentIn F x)

full_name: isOpen_connectedComponent
code: theorem isOpen_connectedComponent [LocallyConnectedSpace α] {x : α} :
    IsOpen (connectedComponent x) 
goal: IsOpen (connectedComponent x)

full_name: isClopen_connectedComponent
code: theorem isClopen_connectedComponent [LocallyConnectedSpace α] {x : α} :
    IsClopen (connectedComponent x) 
goal: IsClopen (connectedComponent x)

full_name: locallyConnectedSpace_iff_connectedComponentIn_open
code: theorem locallyConnectedSpace_iff_connectedComponentIn_open :
    LocallyConnectedSpace α ↔
      ∀ F : Set α, IsOpen F → ∀ x ∈ F, IsOpen (connectedComponentIn F x) 
goal: LocallyConnectedSpace α ↔ ∀ (F : Set α), IsOpen F → ∀ x ∈ F, IsOpen (connectedComponentIn F x)

full_name: locallyConnectedSpace_iff_connected_subsets
code: theorem locallyConnectedSpace_iff_connected_subsets :
    LocallyConnectedSpace α ↔ ∀ (x : α), ∀ U ∈ 𝓝 x, ∃ V ∈ 𝓝 x, IsPreconnected V ∧ V ⊆ U 
goal: LocallyConnectedSpace α ↔ ∀ (x : α), ∀ U ∈ 𝓝 x, ∃ V ∈ 𝓝 x, IsPreconnected V ∧ V ⊆ U

full_name: locallyConnectedSpace_iff_connected_basis
code: theorem locallyConnectedSpace_iff_connected_basis :
    LocallyConnectedSpace α ↔
      ∀ x, (𝓝 x).HasBasis (fun s : Set α => s ∈ 𝓝 x ∧ IsPreconnected s) id 
goal: LocallyConnectedSpace α ↔ ∀ (x : α), (𝓝 x).HasBasis (fun s => s ∈ 𝓝 x ∧ IsPreconnected s) id

full_name: locallyConnectedSpace_of_connected_bases
code: theorem locallyConnectedSpace_of_connected_bases {ι : Type*} (b : α → ι → Set α) (p : α → ι → Prop)
    (hbasis : ∀ x, (𝓝 x).HasBasis (p x) (b x))
    (hconnected : ∀ x i, p x i → IsPreconnected (b x i)) : LocallyConnectedSpace α 
goal: LocallyConnectedSpace α

full_name: OpenEmbedding.locallyConnectedSpace
code: theorem OpenEmbedding.locallyConnectedSpace [LocallyConnectedSpace α] [TopologicalSpace β]
    {f : β → α} (h : OpenEmbedding f) : LocallyConnectedSpace β 
goal: LocallyConnectedSpace β

full_name: IsOpen.locallyConnectedSpace
code: theorem IsOpen.locallyConnectedSpace [LocallyConnectedSpace α] {U : Set α} (hU : IsOpen U) :
    LocallyConnectedSpace U 
goal: LocallyConnectedSpace ↑U