exists_open_prod_subset_of_mem_nhds_diagonal

English

There exists a finite family of nonempty clopen sets that cover the space and are pairwise disjoint.

Русский

Существует конечная семья непустых парно непересекающихся клиноподпространств, покрывающих пространство.

LaTeX

$$$\exists n\exists W: Fin n \to Clopens X,\; \forall j, W_j \neq \emptyset \land ∃ i, W_j\subset U_i \\land univ \subseteq \bigcup_j W_j \\land Pairwise (Disjoint on W).$$$

Lean4

/-- If `S` is any neighbourhood of the diagonal in a topological space `X`, any point of `X` has an
open neighbourhood `U` such that `U ×ˢ U ⊆ S`. -/
theorem exists_open_prod_subset_of_mem_nhds_diagonal (hS : S ∈ nhdsSet (diagonal X)) (x : X) :
    ∃ U : Set X, IsOpen U ∧ x ∈ U ∧ U ×ˢ U ⊆ S :=
  by
  have : S ∈ 𝓝 (x, x) := mem_nhdsSet_iff_forall.mp hS _ rfl
  obtain ⟨u, v, huo, hux, hvo, hvx, H⟩ := by rwa [mem_nhds_prod_iff'] at this
  exact ⟨_, huo.inter hvo, ⟨hux, hvx⟩, fun p hp ↦ H ⟨hp.1.1, hp.2.2⟩⟩

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