Theorems · Theorem · general topology

Pairwise.countable_of_isOpen_disjoint

∀ {α : Type u} [t : TopologicalSpace α] [TopologicalSpace.SeparableSpace α] {ι : Type u_2} {s : ι → Set α},
  Pairwise (Function.onFun Disjoint s) → (∀ (i : ι), IsOpen (s i)) → (∀ (i : ι), (s i).Nonempty) → Countable ι

In a separable space, a family of nonempty disjoint open sets is countable.

Defined in

Mathlib.Topology.Bases

Cited by

1 results in Mathlib

Foundations

Depth 66 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

TopologicalSpaceTopologicalSpace.SeparableSpace