feat(topology/metric_space): introduce is_metric_separated (#6685)

Author: urkud

src/topology/metric_space/metric_separated.lean

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+/-
+Copyright (c) 2021 Yury Kudryashov. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Author: Yury Kudryashov
+-/
+import topology.metric_space.emetric_space
+
+/-!
+# Metric separated pairs of sets
+
+In this file we define the predicate `is_metric_separated`. We say that two sets in an (extended)
+metric space are *metric separated* if the (extended) distance between `x ∈ s` and `y ∈ t` is
+bounded from below by a positive constant.
+
+This notion is useful, e.g., to define metric outer measures.
+-/
+
+open emetric set
+noncomputable theory
+
+/-- Two sets in an (extended) metric space are called *metric separated* if the (extended) distance
+between `x ∈ s` and `y ∈ t` is bounded from below by a positive constant. -/
+def is_metric_separated {X : Type*} [emetric_space X] (s t : set X) :=
+∃ r ≠ 0, ∀ (x ∈ s) (y ∈ t), r ≤ edist x y
+
+namespace is_metric_separated
+
+variables {X : Type*} [emetric_space X] {s t : set X} {x y : X}
+
+@[symm] lemma symm (h : is_metric_separated s t) : is_metric_separated t s :=
+let ⟨r, r0, hr⟩ := h in ⟨r, r0, λ y hy x hx, edist_comm x y ▸ hr x hx y hy⟩
+
+lemma comm : is_metric_separated s t ↔ is_metric_separated t s := ⟨symm, symm⟩
+
+@[simp] lemma empty_left (s : set X) : is_metric_separated ∅ s :=
+⟨1, ennreal.zero_lt_one.ne', λ x, false.elim⟩
+
+@[simp] lemma empty_right (s : set X) : is_metric_separated s ∅ :=
+(empty_left s).symm
+
+protected lemma disjoint (h : is_metric_separated s t) : disjoint s t :=
+let ⟨r, r0, hr⟩ := h in λ x hx, r0 $ by simpa using hr x hx.1 x hx.2
+
+lemma subset_compl_right (h : is_metric_separated s t) : s ⊆ tᶜ :=
+λ x hs ht, h.disjoint ⟨hs, ht⟩
+
+@[mono] lemma mono {s' t'} (hs : s ⊆ s') (ht : t ⊆ t') :
+  is_metric_separated s' t' → is_metric_separated s t :=
+λ ⟨r, r0, hr⟩, ⟨r, r0, λ x hx y hy, hr x (hs hx) y (ht hy)⟩
+
+lemma mono_left {s'} (h' : is_metric_separated s' t) (hs : s ⊆ s') :
+  is_metric_separated s t :=
+h'.mono hs subset.rfl
+
+lemma mono_right {t'} (h' : is_metric_separated s t') (ht : t ⊆ t') :
+  is_metric_separated s t :=
+h'.mono subset.rfl ht
+
+lemma union_left {s'} (h : is_metric_separated s t) (h' : is_metric_separated s' t) :
+  is_metric_separated (s ∪ s') t :=
+begin
+  rcases ⟨h, h'⟩ with ⟨⟨r, r0, hr⟩, ⟨r', r0', hr'⟩⟩,
+  refine ⟨min r r', _, λ x hx y hy, hx.elim _ _⟩,
+  { rw [← pos_iff_ne_zero] at r0 r0' ⊢,
+    exact lt_min r0 r0' },
+  { exact λ hx, (min_le_left _ _).trans (hr _ hx _ hy) },
+  { exact λ hx, (min_le_right _ _).trans (hr' _ hx _ hy) }
+end
+
+@[simp] lemma union_left_iff {s'} :
+  is_metric_separated (s ∪ s') t ↔ is_metric_separated s t ∧ is_metric_separated s' t :=
+⟨λ h, ⟨h.mono_left (subset_union_left _ _), h.mono_left (subset_union_right _ _)⟩,
+  λ h, h.1.union_left h.2⟩
+
+lemma union_right {t'} (h : is_metric_separated s t) (h' : is_metric_separated s t') :
+  is_metric_separated s (t ∪ t') :=
+(h.symm.union_left h'.symm).symm
+
+@[simp] lemma union_right_iff {t'} :
+  is_metric_separated s (t ∪ t') ↔ is_metric_separated s t ∧ is_metric_separated s t' :=
+comm.trans $ union_left_iff.trans $ and_congr comm comm
+
+lemma finite_Union_left_iff {ι : Type*} {I : set ι} (hI : finite I) {s : ι → set X} {t : set X} :
+  is_metric_separated (⋃ i ∈ I, s i) t ↔ ∀ i ∈ I, is_metric_separated (s i) t :=
+begin
+  refine finite.induction_on hI (by simp) (λ i I hi _ hI, _),
+  rw [bUnion_insert, ball_insert_iff, union_left_iff, hI]
+end
+
+alias finite_Union_left_iff ↔ _ is_metric_separated.finite_Union_left
+
+lemma finite_Union_right_iff {ι : Type*} {I : set ι} (hI : finite I) {s : set X} {t : ι → set X} :
+  is_metric_separated s (⋃ i ∈ I, t i) ↔ ∀ i ∈ I, is_metric_separated s (t i) :=
+by simpa only [@comm _ _ s] using finite_Union_left_iff hI
+
+@[simp] lemma finset_Union_left_iff {ι : Type*} {I : finset ι} {s : ι → set X} {t : set X} :
+  is_metric_separated (⋃ i ∈ I, s i) t ↔ ∀ i ∈ I, is_metric_separated (s i) t :=
+finite_Union_left_iff I.finite_to_set
+
+alias finset_Union_left_iff ↔ _ is_metric_separated.finset_Union_left
+
+@[simp] lemma finset_Union_right_iff {ι : Type*} {I : finset ι} {s : set X} {t : ι → set X} :
+  is_metric_separated s (⋃ i ∈ I, t i) ↔ ∀ i ∈ I, is_metric_separated s (t i) :=
+finite_Union_right_iff I.finite_to_set
+
+alias finset_Union_right_iff ↔ _ is_metric_separated.finset_Union_right
+
+end is_metric_separated