hasBasis_nhds_prod — Mathlib

English

Same as isClosed_ball but stated for any x ∈ α and V closed; closed ball set remains closed.

Русский

То же самое, но для любого x и V замкнутого; шар по замкнутому набору остаётся замкнутым.

LaTeX

$$$\\operatorname{IsClosed}(V) \\Rightarrow \\operatorname{IsClosed}(\\mathrm{ball}\\, x\\, V)$$$

Lean4

theorem hasBasis_nhds_prod (x y : α) :
    HasBasis (𝓝 (x, y)) (fun s => s ∈ 𝓤 α ∧ IsSymmetricRel s) fun s => ball x s ×ˢ ball y s :=
  by
  rw [nhds_prod_eq]
  apply (hasBasis_nhds x).prod_same_index (hasBasis_nhds y)
  rintro U V ⟨U_in, U_symm⟩ ⟨V_in, V_symm⟩
  exact ⟨U ∩ V, ⟨(𝓤 α).inter_sets U_in V_in, U_symm.inter V_symm⟩, ball_inter_left x U V, ball_inter_right y U V⟩

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