[Merged by Bors] - feat: port Topology.Instances.Int

Mathlib.lean

@@ -1278,6 +1278,7 @@ import Mathlib.Topology.GDelta
 import Mathlib.Topology.Hom.Open
 import Mathlib.Topology.Homeomorph
 import Mathlib.Topology.Inseparable
+import Mathlib.Topology.Instances.Int
 import Mathlib.Topology.Instances.Sign
 import Mathlib.Topology.IsLocallyHomeomorph
 import Mathlib.Topology.List

Mathlib/Topology/Instances/Int.lean

@@ -0,0 +1,87 @@
+/-
+Copyright (c) 2017 Johannes Hölzl. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Johannes Hölzl, Mario Carneiro
+
+! This file was ported from Lean 3 source module topology.instances.int
+! leanprover-community/mathlib commit 70fd9563a21e7b963887c9360bd29b2393e6225a
+! Please do not edit these lines, except to modify the commit id
+! if you have ported upstream changes.
+-/
+import Mathlib.Data.Int.Interval
+import Mathlib.Topology.MetricSpace.Basic
+import Mathlib.Order.Filter.Archimedean
+
+/-!
+# Topology on the integers
+
+The structure of a metric space on `ℤ` is introduced in this file, induced from `ℝ`.
+-/
+
+
+noncomputable section
+
+open Metric Set Filter
+
+namespace Int
+
+instance : Dist ℤ :=
+  ⟨fun x y => dist (x : ℝ) y⟩
+
+theorem dist_eq (x y : ℤ) : dist x y = |(x : ℝ) - y| := rfl
+#align int.dist_eq Int.dist_eq
+
+theorem dist_eq' (m n : ℤ) : dist m n = |m - n| := by rw [dist_eq]; norm_cast
+
+@[norm_cast, simp]
+theorem dist_cast_real (x y : ℤ) : dist (x : ℝ) y = dist x y :=
+  rfl
+#align int.dist_cast_real Int.dist_cast_real
+
+theorem pairwise_one_le_dist : Pairwise fun m n : ℤ => 1 ≤ dist m n := by
+  intro m n hne
+  rw [dist_eq]; norm_cast; rwa [← zero_add (1 : ℤ), Int.add_one_le_iff, abs_pos, sub_ne_zero]
+#align int.pairwise_one_le_dist Int.pairwise_one_le_dist
+
+theorem uniformEmbedding_coe_real : UniformEmbedding ((↑) : ℤ → ℝ) :=
+  uniformEmbedding_bot_of_pairwise_le_dist zero_lt_one pairwise_one_le_dist
+#align int.uniform_embedding_coe_real Int.uniformEmbedding_coe_real
+
+theorem closedEmbedding_coe_real : ClosedEmbedding ((↑) : ℤ → ℝ) :=
+  closedEmbedding_of_pairwise_le_dist zero_lt_one pairwise_one_le_dist
+#align int.closed_embedding_coe_real Int.closedEmbedding_coe_real
+
+instance : MetricSpace ℤ := Int.uniformEmbedding_coe_real.comapMetricSpace _
+
+theorem preimage_ball (x : ℤ) (r : ℝ) : (↑) ⁻¹' ball (x : ℝ) r = ball x r := rfl
+#align int.preimage_ball Int.preimage_ball
+
+theorem preimage_closedBall (x : ℤ) (r : ℝ) : (↑) ⁻¹' closedBall (x : ℝ) r = closedBall x r := rfl
+#align int.preimage_closed_ball Int.preimage_closedBall
+
+theorem ball_eq_Ioo (x : ℤ) (r : ℝ) : ball x r = Ioo ⌊↑x - r⌋ ⌈↑x + r⌉ := by
+  rw [← preimage_ball, Real.ball_eq_Ioo, preimage_Ioo]
+#align int.ball_eq_Ioo Int.ball_eq_Ioo
+
+theorem closedBall_eq_Icc (x : ℤ) (r : ℝ) : closedBall x r = Icc ⌈↑x - r⌉ ⌊↑x + r⌋ := by
+  rw [← preimage_closedBall, Real.closedBall_eq_Icc, preimage_Icc]
+#align int.closed_ball_eq_Icc Int.closedBall_eq_Icc
+
+instance : ProperSpace ℤ :=
+  ⟨fun x r => by
+    rw [closedBall_eq_Icc]
+    exact (Set.finite_Icc _ _).isCompact⟩
+
+@[simp]
+theorem cocompact_eq : cocompact ℤ = atBot ⊔ atTop := by
+  simp_rw [← comap_dist_right_atTop_eq_cocompact (0 : ℤ), dist_eq', sub_zero,
+    ← comap_abs_atTop, ← @Int.comap_cast_atTop ℝ, comap_comap]; rfl
+#align int.cocompact_eq Int.cocompact_eq
+
+@[simp]
+theorem cofinite_eq : (cofinite : Filter ℤ) = atBot ⊔ atTop := by
+  rw [← cocompact_eq_cofinite, cocompact_eq]
+#align int.cofinite_eq Int.cofinite_eq
+
+end Int
+