English
Let s ⊆ X be compact and f be a filter on Y with f disjoint from cocompact Y. Then 𝓝ˢ s ×ˢ f ≤ 𝓝ˢ (s ×ˢ Set.univ).
Русский
Пусть s ⊆ X — компактно, а фильтр f на Y дисjoint с cocompact Y. Тогда 𝓝ˢ s ×ˢ f вложено в 𝓝ˢ (s ×ˢ Set.univ).
LaTeX
$$$\mathcal{N}_{\mathrm{set}}(s) \times f \le \mathcal{N}_{\mathrm{set}}(s \times \mathrm{univ})$$$
Lean4
theorem nhdsSet_prod_le_of_disjoint_cocompact {f : Filter Y} (hs : IsCompact s) (hf : Disjoint f (Filter.cocompact Y)) :
𝓝ˢ s ×ˢ f ≤ 𝓝ˢ (s ×ˢ Set.univ) :=
by
obtain ⟨K, hKf, hK⟩ := (disjoint_cocompact_right f).mp hf
calc
𝓝ˢ s ×ˢ f
_ ≤ 𝓝ˢ s ×ˢ 𝓟 K := (Filter.prod_mono_right _ (Filter.le_principal_iff.mpr hKf))
_ ≤ 𝓝ˢ s ×ˢ 𝓝ˢ K := (Filter.prod_mono_right _ principal_le_nhdsSet)
_ = 𝓝ˢ (s ×ˢ K) := (hs.nhdsSet_prod_eq hK).symm
_ ≤ 𝓝ˢ (s ×ˢ Set.univ) := nhdsSet_mono (prod_mono_right le_top)
