feat(topology/metric_space): use weaker TC assumptions (#14316)

Assume `t0_space` instead of `separated_space` in `metric.of_t0_pseudo_metric_space` and `emetric.of_t0_pseudo_emetric_space` (both definition used to have `t2` in their names).

src/analysis/normed/group/basic.lean

@@ -1028,18 +1028,18 @@ lemma semi_normed_group.mem_closure_iff {s : set E} {x : E} :
   x ∈ closure s ↔ ∀ ε > 0, ∃ y ∈ s, ∥x - y∥ < ε :=
 by simp [metric.mem_closure_iff, dist_eq_norm]
 
-lemma norm_le_zero_iff' [separated_space E] {g : E} :
+lemma norm_le_zero_iff' [t0_space E] {g : E} :
   ∥g∥ ≤ 0 ↔ g = 0 :=
 begin
-  letI : normed_group E := { to_metric_space := of_t2_pseudo_metric_space ‹_›,
+  letI : normed_group E := { to_metric_space := metric.of_t0_pseudo_metric_space E,
     .. ‹semi_normed_group E› },
   rw [← dist_zero_right], exact dist_le_zero
 end
 
-lemma norm_eq_zero_iff' [separated_space E] {g : E} : ∥g∥ = 0 ↔ g = 0 :=
+lemma norm_eq_zero_iff' [t0_space E] {g : E} : ∥g∥ = 0 ↔ g = 0 :=
 (norm_nonneg g).le_iff_eq.symm.trans norm_le_zero_iff'
 
-lemma norm_pos_iff' [separated_space E] {g : E} : 0 < ∥g∥ ↔ g ≠ 0 :=
+lemma norm_pos_iff' [t0_space E] {g : E} : 0 < ∥g∥ ↔ g ≠ 0 :=
 by rw [← not_le, norm_le_zero_iff']
 
 lemma cauchy_seq_sum_of_eventually_eq {u v : ℕ → E} {N : ℕ} (huv : ∀ n ≥ N, u n = v n)

src/topology/metric_space/basic.lean

@@ -207,7 +207,7 @@ def pseudo_metric_space.of_metrizable {α : Type*} [topological_space α] (dist
   (dist_comm : ∀ x y : α, dist x y = dist y x)
   (dist_triangle : ∀ x y z : α, dist x z ≤ dist x y + dist y z)
   (H : ∀ s : set α, is_open s ↔ ∀ x ∈ s, ∃ ε > 0, ∀ y, dist x y < ε → y ∈ s) :
-pseudo_metric_space α :=
+  pseudo_metric_space α :=
 { dist := dist,
   dist_self := dist_self,
   dist_comm := dist_comm,
@@ -1022,6 +1022,10 @@ by { convert metric.emetric_closed_ball ε.2, simp }
 @[simp] lemma metric.emetric_ball_top (x : α) : emetric.ball x ⊤ = univ :=
 eq_univ_of_forall $ λ y, edist_lt_top _ _
 
+lemma metric.indistinguishable_iff {x y : α} : indistinguishable x y ↔ dist x y = 0 :=
+by rw [emetric.indistinguishable_iff, edist_nndist, dist_nndist, ennreal.coe_eq_zero,
+  nnreal.coe_eq_zero]
+
 /-- Build a new pseudometric space from an old one where the bundled uniform structure is provably
 (but typically non-definitionaly) equal to some given uniform structure.
 See Note [forgetful inheritance].
@@ -2475,26 +2479,20 @@ begin
 end
 
 @[priority 100] -- see Note [lower instance priority]
-instance metric_space.to_separated : separated_space γ :=
+instance _root_.metric_space.to_separated : separated_space γ :=
 separated_def.2 $ λ x y h, eq_of_forall_dist_le $
   λ ε ε0, le_of_lt (h _ (dist_mem_uniformity ε0))
 
-/-- If a `pseudo_metric_space` is separated, then it is a `metric_space`. -/
-def of_t2_pseudo_metric_space {α : Type*} [pseudo_metric_space α]
-  (h : separated_space α) : metric_space α :=
-{ eq_of_dist_eq_zero := λ x y hdist,
-  begin
-    refine separated_def.1 h x y (λ s hs, _),
-    obtain ⟨ε, hε, H⟩ := mem_uniformity_dist.1 hs,
-    exact H (show dist x y < ε, by rwa [hdist])
-  end
+/-- If a `pseudo_metric_space` is a T₀ space, then it is a `metric_space`. -/
+def of_t0_pseudo_metric_space (α : Type*) [pseudo_metric_space α] [t0_space α] :
+  metric_space α :=
+{ eq_of_dist_eq_zero := λ x y hdist, indistinguishable.eq $ metric.indistinguishable_iff.2 hdist,
   ..‹pseudo_metric_space α› }
 
 /-- A metric space induces an emetric space -/
 @[priority 100] -- see Note [lower instance priority]
 instance metric_space.to_emetric_space : emetric_space γ :=
-{ eq_of_edist_eq_zero := assume x y h, by simpa [edist_dist] using h,
-  ..pseudo_metric_space.to_pseudo_emetric_space, }
+emetric.of_t0_pseudo_emetric_space γ
 
 lemma is_closed_of_pairwise_le_dist {s : set γ} {ε : ℝ} (hε : 0 < ε)
   (hs : s.pairwise (λ x y, ε ≤ dist x y)) : is_closed s :=

src/topology/metric_space/emetric_space.lean

@@ -613,6 +613,9 @@ theorem tendsto_at_top [nonempty β] [semilattice_sup β] {u : β → α} {a : 
 (at_top_basis.tendsto_iff nhds_basis_eball).trans $
   by simp only [exists_prop, true_and, mem_Ici, mem_ball]
 
+theorem indistinguishable_iff : indistinguishable x y ↔ edist x y = 0 :=
+by simp [indistinguishable_iff_closure, mem_closure_iff, edist_comm, forall_lt_iff_le']
+
 /-- In a pseudoemetric space, Cauchy sequences are characterized by the fact that, eventually,
 the pseudoedistance between its elements is arbitrarily small -/
 @[nolint ge_or_gt] -- see Note [nolint_ge]
@@ -698,6 +701,7 @@ to avoid a loop with `sigma_compact_space_of_locally_compact_second_countable`.
 lemma second_countable_of_sigma_compact [sigma_compact_space α] :
   second_countable_topology α :=
 begin
+
   suffices : separable_space α, by exactI uniform_space.second_countable_of_separable α,
   choose T hTsub hTc hsubT
     using λ n, subset_countable_closure_of_compact (is_compact_compact_covering α n),
@@ -883,15 +887,10 @@ instance to_separated : separated_space γ :=
 separated_def.2 $ λ x y h, eq_of_forall_edist_le $
 λ ε ε0, le_of_lt (h _ (edist_mem_uniformity ε0))
 
-/-- If a  `pseudo_emetric_space` is separated, then it is an `emetric_space`. -/
-def emetric_of_t2_pseudo_emetric_space {α : Type*} [pseudo_emetric_space α]
-  (h : separated_space α) : emetric_space α :=
-{ eq_of_edist_eq_zero := λ x y hdist,
-  begin
-    refine separated_def.1 h x y (λ s hs, _),
-    obtain ⟨ε, hε, H⟩ := mem_uniformity_edist.1 hs,
-    exact H (show edist x y < ε, by rwa [hdist])
-  end
+/-- If a `pseudo_emetric_space` is a T₀ space, then it is an `emetric_space`. -/
+def emetric.of_t0_pseudo_emetric_space (α : Type*) [pseudo_emetric_space α] [t0_space α] :
+  emetric_space α :=
+{ eq_of_edist_eq_zero := λ x y hdist, indistinguishable.eq $ emetric.indistinguishable_iff.2 hdist,
   ..‹pseudo_emetric_space α› }
 
 /-- Auxiliary function to replace the uniformity on an emetric space with
@@ -998,12 +997,7 @@ lemma diam_eq_zero_iff : diam s = 0 ↔ s.subsingleton :=
 ⟨λ h x hx y hy, edist_le_zero.1 $ h ▸ edist_le_diam_of_mem hx hy, diam_subsingleton⟩
 
 lemma diam_pos_iff : 0 < diam s ↔ ∃ (x ∈ s) (y ∈ s), x ≠ y :=
-begin
-  have := not_congr (@diam_eq_zero_iff _ _ s),
-  dunfold set.subsingleton at this,
-  push_neg at this,
-  simpa only [pos_iff_ne_zero, exists_prop] using this
-end
+by simp only [pos_iff_ne_zero, ne.def, diam_eq_zero_iff, set.subsingleton, not_forall]
 
 end diam