English
For any X, T35Space X iff stoneCechUnit is an embedding.
Русский
Для любого X пространство T35 тогда и только тогда, когда stoneCechUnit является вложением.
LaTeX
$$$T35Space X \iff IsEmbedding(\mathrm{stoneCechUnit}: X \to \mathrm{StoneCech} X)$$$
Lean4
/-- If `X` is a topological space, `s` is a set in `X` such that the induced topology is T₀ and is
second countable, then there exists a countable family of open sets in `X` that separates points
of `s`. -/
instance [TopologicalSpace X] {s : Set X} [T0Space s] [SecondCountableTopology s] :
HasCountableSeparatingOn X IsOpen s :=
by
suffices HasCountableSeparatingOn s IsOpen univ from .of_subtype fun _ ↦ isOpen_induced_iff.1
refine ⟨⟨countableBasis s, countable_countableBasis _, fun _ ↦ isOpen_of_mem_countableBasis, fun x _ y _ h ↦ ?_⟩⟩
exact ((isBasis_countableBasis _).inseparable_iff.2 h).eq
