locPathConnectedSpace — Mathlib

English

Disjoint unions of locally path-connected spaces indexed by i are locally path-connected.

Русский

Дисjoint объединения локально путевых пространств по индексу i локально путеподключены.

LaTeX

$$LocPathConnectedSpace (Σ i, X_i)$$

Lean4

/-- Disjoint unions of locally path-connected spaces are locally path-connected. -/
instance locPathConnectedSpace {X : ι → Type*} [(i : ι) → TopologicalSpace (X i)]
    [(i : ι) → LocPathConnectedSpace (X i)] : LocPathConnectedSpace ((i : ι) × X i) :=
  by
  rw [locPathConnectedSpace_iff_pathComponentIn_mem_nhds]; intro x u hu hxu; rw [mem_nhds_iff]
  refine ⟨(Sigma.mk x.1) '' (pathComponentIn ((Sigma.mk x.1) ⁻¹' u) x.2), ?_, ?_, ?_⟩
  · apply IsPathConnected.subset_pathComponentIn
    · exact (isPathConnected_pathComponentIn (by exact hxu)).image continuous_sigmaMk
    · exact ⟨x.2, mem_pathComponentIn_self hxu, rfl⟩
    · exact (image_mono pathComponentIn_subset).trans (u.image_preimage_subset _)
  · exact isOpenMap_sigmaMk _ <| (hu.preimage continuous_sigmaMk).pathComponentIn _
  · exact ⟨x.2, mem_pathComponentIn_self hxu, rfl⟩

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