Theorems · Theorem · general topology

Set.PairwiseDisjoint.countable_of_isOpen

∀ {α : Type u} [t : TopologicalSpace α] [TopologicalSpace.SeparableSpace α] {ι : Type u_2} {s : ι → Set α} {a : Set ι},
  a.PairwiseDisjoint s → (∀ i ∈ a, IsOpen (s i)) → (∀ i ∈ a, (s i).Nonempty) → a.Countable

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

Defined in

Mathlib.Topology.Bases

Cited by

4 results in Mathlib

Foundations

Depth 67 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

TopologicalSpaceTopologicalSpace.SeparableSpace