dynamics.fixed_points.topology
⟷
Mathlib.Dynamics.FixedPoints.Topology
The following section lists changes to this file in mathlib3 and mathlib4 that occured after the initial port. Most recent changes are shown first. Hovering over a commit will show all commits associated with the same mathlib3 commit.
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mathlib commit https://github.com/leanprover-community/mathlib/commit/ce64cd319bb6b3e82f31c2d38e79080d377be451
@@ -3,8 +3,8 @@ Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Johannes Hölzl
-/
-import Mathbin.Dynamics.FixedPoints.Basic
-import Mathbin.Topology.Separation
+import Dynamics.FixedPoints.Basic
+import Topology.Separation
#align_import dynamics.fixed_points.topology from "leanprover-community/mathlib"@"4c19a16e4b705bf135cf9a80ac18fcc99c438514"
mathlib commit https://github.com/leanprover-community/mathlib/commit/8ea5598db6caeddde6cb734aa179cc2408dbd345
@@ -2,15 +2,12 @@
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Johannes Hölzl
-
-! This file was ported from Lean 3 source module dynamics.fixed_points.topology
-! leanprover-community/mathlib commit 4c19a16e4b705bf135cf9a80ac18fcc99c438514
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
-/
import Mathbin.Dynamics.FixedPoints.Basic
import Mathbin.Topology.Separation
+#align_import dynamics.fixed_points.topology from "leanprover-community/mathlib"@"4c19a16e4b705bf135cf9a80ac18fcc99c438514"
+
/-!
# Topological properties of fixed points
mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb
@@ -32,7 +32,7 @@ variable {α : Type _} [TopologicalSpace α] [T2Space α] {f : α → α}
open Function Filter
-open Topology
+open scoped Topology
#print isFixedPt_of_tendsto_iterate /-
/-- If the iterates `f^[n] x` converge to `y` and `f` is continuous at `y`,
mathlib commit https://github.com/leanprover-community/mathlib/commit/bd9851ca476957ea4549eb19b40e7b5ade9428cc
Type _
and Sort _
(#6499)
We remove all possible occurences of Type _
and Sort _
in favor of Type*
and Sort*
.
This has nice performance benefits.
@@ -22,7 +22,7 @@ fixed points, iterates
-/
-variable {α : Type _} [TopologicalSpace α] [T2Space α] {f : α → α}
+variable {α : Type*} [TopologicalSpace α] [T2Space α] {f : α → α}
open Function Filter
@@ -2,15 +2,12 @@
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Johannes Hölzl
-
-! This file was ported from Lean 3 source module dynamics.fixed_points.topology
-! leanprover-community/mathlib commit d90e4e186f1d18e375dcd4e5b5f6364b01cb3e46
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
-/
import Mathlib.Dynamics.FixedPoints.Basic
import Mathlib.Topology.Separation
+#align_import dynamics.fixed_points.topology from "leanprover-community/mathlib"@"d90e4e186f1d18e375dcd4e5b5f6364b01cb3e46"
+
/-!
# Topological properties of fixed points
@@ -33,7 +33,7 @@ open Topology
/-- If the iterates `f^[n] x` converge to `y` and `f` is continuous at `y`,
then `y` is a fixed point for `f`. -/
-theorem isFixedPt_of_tendsto_iterate {x y : α} (hy : Tendsto (fun n => (f^[n]) x) atTop (𝓝 y))
+theorem isFixedPt_of_tendsto_iterate {x y : α} (hy : Tendsto (fun n => f^[n] x) atTop (𝓝 y))
(hf : ContinuousAt f y) : IsFixedPt f y := by
refine' tendsto_nhds_unique ((tendsto_add_atTop_iff_nat 1).1 _) hy
simp only [iterate_succ' f]
@@ -44,4 +44,3 @@ theorem isFixedPt_of_tendsto_iterate {x y : α} (hy : Tendsto (fun n => (f^[n])
theorem isClosed_fixedPoints (hf : Continuous f) : IsClosed (fixedPoints f) :=
isClosed_eq hf continuous_id
#align is_closed_fixed_points isClosed_fixedPoints
-
The unported dependencies are