English
If a family of nonempty open disjoint sets covers a preconnected space, then at most one of them can be nonempty; more generally, any disjoint family of nonempty clopen subsets is a singleton.
Русский
Если непустые открытые попарно не пересекающиеся множества покрывают предпредсвязанное пространство, то таких множеств может быть не более одного; в общем случае дисjoint семейство непустых открытых является единичным.
LaTeX
$$$\text{Disjoint}(s_i) \Longrightarrow \text{ Subsingleton } i$ under preconnectedness; more precisely, for disjoint clopen family, at most one index is nonempty.$$
Lean4
/-- In a preconnected space, any disjoint family of non-empty clopen subsets has at most one
element. -/
theorem subsingleton_of_disjoint_isClopen (h_clopen : ∀ i, IsClopen (s i)) : Subsingleton ι :=
by
replace h_nonempty : ∀ i, s i ≠ ∅ := by intro i; rw [← nonempty_iff_ne_empty]; exact h_nonempty i
rw [← not_nontrivial_iff_subsingleton]
by_contra contra
obtain ⟨i, j, h_ne⟩ := contra
replace h_ne : s i ∩ s j = ∅ := by
simpa only [← bot_eq_empty, eq_bot_iff, ← inf_eq_inter, ← disjoint_iff_inf_le] using h_disj h_ne
rcases isClopen_iff.mp (h_clopen i) with hi | hi
· exact h_nonempty i hi
· rw [hi, univ_inter] at h_ne
exact h_nonempty j h_ne
