English
A T0 space with a clopen basis is totally separated.
Русский
У T0-пространства с базисом из открыто-перекрывающих множеств клипповых множеств существует полное разделение.
LaTeX
$$$\\text{If } X \\text{ is T0 and has a basis of clopen sets, then } X \\text{ is TotallySeparated}$.$$
Lean4
/-- Let `s` be a closed set in a separable normal space. If the induced topology on `s` is discrete,
then `s` has cardinality less than continuum.
The proof follows
https://en.wikipedia.org/wiki/Moore_plane#Proof_that_the_Moore_plane_is_not_normal -/
theorem mk_lt_continuum [NormalSpace X] {s : Set X} (hs : IsClosed s) [DiscreteTopology s] : #s < 𝔠 := by
-- Proof by contradiction: assume `𝔠 ≤ #s`
by_contra! h
rcases exists_countable_dense X with ⟨t, htc, htd⟩
haveI := htc.to_subtype
refine (Cardinal.cantor 𝔠).not_ge ?_
calc
-- Any function `s → ℝ` is continuous, hence `2 ^ 𝔠 ≤ #C(s, ℝ)`
2 ^ 𝔠 ≤ #C(s, ℝ) :=
by
rw [ContinuousMap.equivFnOfDiscrete.cardinal_eq, mk_arrow, mk_real, lift_continuum, lift_uzero]
exact
(power_le_power_left two_ne_zero h).trans
(power_le_power_right (nat_lt_continuum 2).le)
-- By the Tietze Extension Theorem, any function `f : C(s, ℝ)` can be extended to `C(X, ℝ)`,
-- hence `#C(s, ℝ) ≤ #C(X, ℝ)`
_ ≤ #C(X, ℝ) := by
choose f hf using ContinuousMap.exists_restrict_eq (Y := ℝ) hs
have hfi : Injective f := LeftInverse.injective hf
exact mk_le_of_injective hfi
_ ≤ #C(t, ℝ) := (mk_le_of_injective <| ContinuousMap.injective_restrict htd)
_ ≤ #(t → ℝ) := (mk_le_of_injective DFunLike.coe_injective)
_ ≤ 𝔠 := by
rw [mk_arrow, mk_real, lift_uzero, lift_continuum, continuum, ← power_mul]
exact power_le_power_left two_ne_zero mk_le_aleph0
