English
A filter l with the countable intersection property is disjoint from the neighborhood filter of a Lindelöf set s if and only if it is disjoint from the neighborhood filter of every point in s.
Русский
Фильтр с CPIP дисjointен с NHDS(s) тогда и только тогда, когда он дисjointен с NHDS(x) для каждого x∈s.
LaTeX
$$$Disjoint\ l\ (\mathcal{N} \!\!\! (s)) \iff \forall x\in s,\ Disjoint\ l\ (\mathcal{N} x).$$$
Lean4
/-- A filter `l` with the countable intersection property is disjoint with the neighborhood
filter of a Lindelöf set if and only if it is disjoint with the neighborhood filter of each point
of this set. -/
theorem disjoint_nhdsSet_right {l : Filter X} [CountableInterFilter l] (hs : IsLindelof s) :
Disjoint l (𝓝ˢ s) ↔ ∀ x ∈ s, Disjoint l (𝓝 x) := by simpa only [disjoint_comm] using hs.disjoint_nhdsSet_left
