English
In a compact space X, the intersection of any family of closed sets with the finite intersection property is nonempty.
Русский
В компактном пространстве X пересечение любойFamily закрытых множеств с конечной свойством пересечений непусто.
LaTeX
$$$\bigcap_{i} t_i \neq \emptyset$ under Finite Intersection Property with each t_i closed and X compact$$
Lean4
/-- Given a family of closed sets `t i` in a compact space, if they satisfy the Finite Intersection
Property, then the intersection of all `t i` is nonempty.
-/
theorem iInter_nonempty {ι : Type v} [CompactSpace X] {t : ι → Set X} (htc : ∀ i, IsClosed (t i))
(hst : ∀ s : Finset ι, (⋂ i ∈ s, t i).Nonempty) : (⋂ i, t i).Nonempty := by
simpa using IsCompact.inter_iInter_nonempty isCompact_univ t htc (by simpa using hst)
