prod_nhdsSet_le_of_disjoint_cocompact

English

Let t ⊆ X be compact and f be a filter on X disjoint from cocompact X. Then f ×ˢ 𝓝ˢ t ≤ 𝓝ˢ (Set.univ ×ˢ t).

Русский

Пусть t ⊆ X — компактно, а фильтр f на X дисjoint с cocompact X. Тогда f ×ˢ 𝓝ˢ t ≤ 𝓝ˢ (univ ×ˢ t).

LaTeX

$$$ f \timesˢ 𝓝ˢ t \le 𝓝ˢ (Set.univ \timesˢ t)$$$

Lean4

theorem prod_nhdsSet_le_of_disjoint_cocompact {t : Set Y} {f : Filter X} (ht : IsCompact t)
    (hf : Disjoint f (Filter.cocompact X)) : f ×ˢ 𝓝ˢ t ≤ 𝓝ˢ (Set.univ ×ˢ t) :=
  by
  obtain ⟨K, hKf, hK⟩ := (disjoint_cocompact_right f).mp hf
  calc
    f ×ˢ 𝓝ˢ t
    _ ≤ (𝓟 K) ×ˢ 𝓝ˢ t := (Filter.prod_mono_left _ (Filter.le_principal_iff.mpr hKf))
    _ ≤ 𝓝ˢ K ×ˢ 𝓝ˢ t := (Filter.prod_mono_left _ principal_le_nhdsSet)
    _ = 𝓝ˢ (K ×ˢ t) := (hK.nhdsSet_prod_eq ht).symm
    _ ≤ 𝓝ˢ (Set.univ ×ˢ t) := nhdsSet_mono (prod_mono_left le_top)

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