/-- In a second countable space, the type of nonempty compact subsets is second countable -/
instance NonemptyCompacts.secondCountableTopology [SecondCountableTopology α] :
    SecondCountableTopology (NonemptyCompacts α) :=
  haveI : SeparableSpace (NonemptyCompacts α) := by
    /- To obtain a countable dense subset of `NonemptyCompacts α`, start from
        a countable dense subset `s` of α, and then consider all its finite nonempty subsets.
        This set is countable and made of nonempty compact sets. It turns out to be dense:
        by total boundedness, any compact set `t` can be covered by finitely many small balls, and
        approximations in `s` of the centers of these balls give the required finite approximation
        of `t`. -/
    rcases exists_countable_dense α with ⟨s, cs, s_dense⟩
    let v0 := { t : Set α | t.Finite ∧ t ⊆ s }
    let v : Set (NonemptyCompacts α) := { t : NonemptyCompacts α | (t : Set α) ∈ v0 }
    refine' ⟨⟨v, _, _⟩⟩
    · have : v0.Countable := countable_setOf_finite_subset cs
      exact this.preimage SetLike.coe_injective
    · refine' fun t => mem_closure_iff.2 fun ε εpos => _
      -- t is a compact nonempty set, that we have to approximate uniformly by a a set in `v`.
      rcases exists_between εpos with ⟨δ, δpos, δlt⟩
      have δpos' : 0 < δ / 2 := ENNReal.half_pos δpos.ne'
      -- construct a map F associating to a point in α an approximating point in s, up to δ/2.
      have Exy : ∀ x, ∃ y, y ∈ s ∧ edist x y < δ / 2 := by
        intro x
        rcases mem_closure_iff.1 (s_dense x) (δ / 2) δpos' with ⟨y, ys, hy⟩
        exact ⟨y, ⟨ys, hy⟩⟩
      let F x := choose (Exy x)
      have Fspec : ∀ x, F x ∈ s ∧ edist x (F x) < δ / 2 := fun x => choose_spec (Exy x)
      -- cover `t` with finitely many balls. Their centers form a set `a`
      have : TotallyBounded (t : Set α) := t.isCompact.totallyBounded
      rcases totallyBounded_iff.1 this (δ / 2) δpos' with ⟨a, af, ta⟩
      -- a : set α, af : a.finite, ta : t ⊆ ⋃ (y : α) (H : y ∈ a), eball y (δ / 2)
      -- replace each center by a nearby approximation in `s`, giving a new set `b`
      let b := F '' a
      have : b.Finite := af.image _
      have tb : ∀ x ∈ t, ∃ y ∈ b, edist x y < δ := by
        intro x hx
        rcases mem_iUnion₂.1 (ta hx) with ⟨z, za, Dxz⟩
        exists F z, mem_image_of_mem _ za
        calc
          edist x (F z) ≤ edist x z + edist z (F z) := edist_triangle _ _ _
          _ < δ / 2 + δ / 2 := (ENNReal.add_lt_add Dxz (Fspec z).2)
          _ = δ := ENNReal.add_halves _
      -- keep only the points in `b` that are close to point in `t`, yielding a new set `c`
      let c := { y ∈ b | ∃ x ∈ t, edist x y < δ }
      have : c.Finite := ‹b.Finite›.subset fun x hx => hx.1
      -- points in `t` are well approximated by points in `c`
      have tc : ∀ x ∈ t, ∃ y ∈ c, edist x y ≤ δ := by
        intro x hx
        rcases tb x hx with ⟨y, yv, Dxy⟩
        have : y ∈ c := by simp [-mem_image]; exact ⟨yv, ⟨x, hx, Dxy⟩⟩
        exact ⟨y, this, le_of_lt Dxy⟩
      -- points in `c` are well approximated by points in `t`
      have ct : ∀ y ∈ c, ∃ x ∈ t, edist y x ≤ δ := by
        rintro y ⟨_, x, xt, Dyx⟩
        have : edist y x ≤ δ :=
          calc
            edist y x = edist x y := edist_comm _ _
            _ ≤ δ := le_of_lt Dyx
        exact ⟨x, xt, this⟩
      -- it follows that their Hausdorff distance is small
      have : hausdorffEdist (t : Set α) c ≤ δ := hausdorffEdist_le_of_mem_edist tc ct
      have Dtc : hausdorffEdist (t : Set α) c < ε := this.trans_lt δlt
      -- the set `c` is not empty, as it is well approximated by a nonempty set
      have hc : c.Nonempty := nonempty_of_hausdorffEdist_ne_top t.nonempty (ne_top_of_lt Dtc)
      -- let `d` be the version of `c` in the type `NonemptyCompacts α`
      let d : NonemptyCompacts α := ⟨⟨c, ‹c.Finite›.isCompact⟩, hc⟩
      have : c ⊆ s := by
        intro x hx
        rcases (mem_image _ _ _).1 hx.1 with ⟨y, ⟨_, yx⟩⟩
        rw [← yx]
        exact (Fspec y).1
      have : d ∈ v := ⟨‹c.Finite›, this⟩
      -- we have proved that `d` is a good approximation of `t` as requested
      

Proof state:
case intro.intro.refine'_2.intro.intro.intro.intro
α : Type u
inst✝¹ : EMetricSpace α
s✝ : Set α
inst✝ : SecondCountableTopology α
s : Set α
cs : Set.Countable s
s_dense : Dense s
v0 : Set (Set α) := {t | Set.Finite t ∧ t ⊆ s}
v : Set (NonemptyCompacts α) := {t | ↑t ∈ v0}
t : NonemptyCompacts α
ε : ℝ≥0∞
εpos : ε > 0
δ : ℝ≥0∞
δpos : 0 < δ
δlt : δ < ε
δpos' : 0 < δ / 2
Exy : ∀ (x : α), ∃ y ∈ s, edist x y < δ / 2
F : α → α := fun x => choose (_ : ∃ y ∈ s, edist x y < δ / 2)
Fspec : ∀ (x : α), F x ∈ s ∧ edist x (F x) < δ / 2
this✝⁴ : TotallyBounded ↑t
a : Set α
af : Set.Finite a
ta : ↑t ⊆ ⋃ y ∈ a, ball y (δ / 2)
b : Set α := F '' a
this✝³ : Set.Finite b
tb : ∀ x ∈ t, ∃ y ∈ b, edist x y < δ
c : Set α := {y | y ∈ b ∧ ∃ x ∈ t, edist x y < δ}
this✝² : Set.Finite c
tc : ∀ x ∈ t, ∃ y ∈ c, edist x y ≤ δ
ct : ∀ y ∈ c, ∃ x ∈ t, edist y x ≤ δ
this✝¹ : hausdorffEdist (↑t) c ≤ δ
Dtc : hausdorffEdist (↑t) c < ε
hc : Set.Nonempty c
d : NonemptyCompacts α := { toCompacts := { carrier := c, isCompact' := (_ : IsCompact c) }, nonempty' := hc }
this✝ : c ⊆ s
this : d ∈ v
⊢ ∃ y ∈ v, edist t y < ε

Next tactic:
exact ⟨d, ‹d ∈ v›, Dtc⟩