Skip to content
This repository was archived by the owner on Jul 24, 2024. It is now read-only.

Commit 2198d2c

Browse files
rwbartonkim-emjcommelinChrisHughes24mergify[bot]
authored
feat(roadmap): add some formal roadmaps in topology (#1914)
* feat(roadmap): add some formal roadmaps in topology * Update roadmap/topology/paracompact.lean Co-Authored-By: Johan Commelin <johan@commelin.net> * Update roadmap/todo.lean * Update roadmap/topology/shrinking_lemma.lean Co-Authored-By: Chris Hughes <33847686+ChrisHughes24@users.noreply.github.com> * add `todo` tactic as a wrapper for `exact todo` * Update roadmap/topology/shrinking_lemma.lean Co-Authored-By: Johan Commelin <johan@commelin.net> * copyright notices and module docs * oops Co-authored-by: Scott Morrison <scott@tqft.net> Co-authored-by: Johan Commelin <johan@commelin.net> Co-authored-by: Chris Hughes <33847686+ChrisHughes24@users.noreply.github.com> Co-authored-by: mergify[bot] <37929162+mergify[bot]@users.noreply.github.com>
1 parent 0c2dbd5 commit 2198d2c

File tree

3 files changed

+157
-0
lines changed

3 files changed

+157
-0
lines changed

‎roadmap/todo.lean

Lines changed: 30 additions & 0 deletions
Original file line numberDiff line numberDiff line change
@@ -0,0 +1,30 @@
1+
/-
2+
Copyright (c) 2020 Reid Barton. All rights reserved.
3+
Released under Apache 2.0 license as described in the file LICENSE.
4+
Authors: Reid Barton
5+
-/
6+
7+
/-!
8+
This file adds an axiom `todo`, and a corresponding tactic,
9+
which can be used in place of `sorry`.
10+
It is only intended for use inside the roadmap subdirectory.
11+
-/
12+
13+
/--
14+
Axiom used to skip proofs in formal roadmaps.
15+
(When working on a roadmap, you may prefer to prove new lemmas,
16+
rather than trying to solve an `exact todo` in-line.
17+
The tactic `extract_goal` is useful for this.)
18+
-/
19+
axiom todo {p : Prop} : p
20+
21+
namespace tactic
22+
namespace interactive
23+
24+
/--
25+
An axiomatic alternative to `sorry`, used in formal roadmaps.
26+
-/
27+
meta def todo : tactic unit := `[exact todo]
28+
29+
end interactive
30+
end tactic

‎roadmap/topology/paracompact.lean

Lines changed: 82 additions & 0 deletions
Original file line numberDiff line numberDiff line change
@@ -0,0 +1,82 @@
1+
/-
2+
Copyright (c) 2020 Reid Barton. All rights reserved.
3+
Released under Apache 2.0 license as described in the file LICENSE.
4+
Authors: Reid Barton
5+
-/
6+
7+
import roadmap.todo
8+
import topology.subset_properties
9+
import topology.separation
10+
import topology.metric_space.basic
11+
12+
/-!
13+
A formal roadmap for basic properties of paracompact spaces.
14+
15+
It contains the statements that compact spaces and metric spaces are paracompact,
16+
and that paracompact t2 spaces are normal, as well as partially formalised proofs.
17+
18+
Any contributor should feel welcome to contribute complete proofs. When this happens,
19+
we should also consider preserving the current file as an exemplar of a formal roadmap.
20+
-/
21+
22+
open set filter
23+
24+
universe u
25+
26+
class paracompact_space (X : Type u) [topological_space X] : Prop :=
27+
(locally_finite_refinement :
28+
∀ {α : Type u} (u : α → set X) (uo : ∀ a, is_open (u a)) (uc : Union u = univ),
29+
∃ {β : Type u} (v : β → set X) (vo : ∀ b, is_open (v b)) (vc : Union v = univ),
30+
locally_finite v ∧ ∀ b, ∃ a, v b ⊆ u a)
31+
32+
/-- Any open cover of a paracompact space has a locally finite *precise* refinement, that is,
33+
one indexed on the same type with each open set contained in the corresponding original one. -/
34+
lemma paracompact_space.precise_refinement {X : Type u} [topological_space X] [paracompact_space X]
35+
{α : Type u} (u : α → set X) (uo : ∀ a, is_open (u a)) (uc : Union u = univ) :
36+
∃ v : α → set X, (∀ a, is_open (v a)) ∧ Union v = univ ∧ locally_finite v ∧ (∀ a, v a ⊆ u a) :=
37+
begin
38+
obtain ⟨β, w, wo, wc, lfw, wr⟩ := paracompact_space.locally_finite_refinement u uo uc,
39+
choose f hf using wr,
40+
refine ⟨λ a, ⋃₀ {s | ∃ b, f b = a ∧ s = w b}, λ a, _, _, _, λ a, _⟩,
41+
{ apply is_open_sUnion _,
42+
rintros t ⟨b, rfl, rfl⟩,
43+
apply wo },
44+
{ todo },
45+
{ todo },
46+
{ apply sUnion_subset,
47+
rintros t ⟨b, rfl, rfl⟩,
48+
apply hf }
49+
end
50+
51+
lemma paracompact_of_compact {X : Type u} [topological_space X] [compact_space X] :
52+
paracompact_space X :=
53+
begin
54+
refine ⟨λ α u uo uc, _⟩,
55+
obtain ⟨s, _, sf, sc⟩ :=
56+
compact_univ.elim_finite_subcover_image (λ a _, uo a) (by rwa [univ_subset_iff, bUnion_univ]),
57+
refine ⟨s, λ b, u b.val, λ b, uo b.val, _, _, λ b, ⟨b.val, subset.refl _⟩⟩,
58+
{ todo },
59+
{ intro x,
60+
refine ⟨univ, univ_mem_sets, _⟩,
61+
todo },
62+
end
63+
64+
lemma normal_of_paracompact_t2 {X : Type u} [topological_space X] [t2_space X]
65+
[paracompact_space X] : normal_space X :=
66+
todo
67+
/-
68+
Similar to the proof of `generalized_tube_lemma`, but different enough not to merge them.
69+
Lemma: if `s : set X` is closed and can be separated from any point by open sets,
70+
then `s` can also be separated from any closed set by open sets. Apply twice.
71+
72+
See
73+
* Bourbaki, General Topology, Chapter IX, §4.4
74+
* https://ncatlab.org/nlab/show/paracompact+Hausdorff+spaces+are+normal
75+
-/
76+
77+
lemma paracompact_of_metric {X : Type u} [metric_space X] : paracompact_space X :=
78+
todo
79+
/-
80+
See Mary Ellen Rudin, A new proof that metric spaces are paracompact.
81+
https://www.ams.org/journals/proc/1969-020-02/S0002-9939-1969-0236876-3/S0002-9939-1969-0236876-3.pdf
82+
-/

‎roadmap/topology/shrinking_lemma.lean

Lines changed: 45 additions & 0 deletions
Original file line numberDiff line numberDiff line change
@@ -0,0 +1,45 @@
1+
/-
2+
Copyright (c) 2020 Reid Barton. All rights reserved.
3+
Released under Apache 2.0 license as described in the file LICENSE.
4+
Authors: Reid Barton
5+
-/
6+
7+
/-!
8+
A formal roadmap for the shrinking lemma for local finite countable covers.
9+
10+
It contains the statement of the lemma, and an informal sketch of the proof,
11+
along with references.
12+
13+
Any contributor should feel welcome to contribute a formal proof. When this happens,
14+
we should also consider preserving the current file as an exemplar of a formal roadmap.
15+
-/
16+
17+
import roadmap.todo
18+
import topology.separation
19+
20+
open set
21+
22+
universes u v
23+
24+
/-- A point-finite open cover of a closed subset of a normal space can be "shrunk" to a new open cover
25+
so that the closure of each new open set is contained in the corresponding original open set. -/
26+
lemma shrinking_lemma {X : Type u} [topological_space X] [normal_space X]
27+
{s : set X} (hs : is_closed s) {α : Type v} (u : α → set X) (uo : ∀ a, is_open (u a))
28+
(uf : ∀ x, finite {a | x ∈ u a}) (su : s ⊆ Union u) :
29+
∃ v : α → set X, s ⊆ Union v ∧ ∀ a, is_open (v a) ∧ closure (v a) ⊆ u a :=
30+
todo
31+
/-
32+
Apply Zorn's lemma to
33+
T = Σ (i : set α), {v : α → set X // s ⊆ Union v ∧ (∀ a, is_open (v a)) ∧
34+
(∀ a ∈ i, closure (v a) ⊆ u a) ∧ (∀ a ∉ i, v a = u a)}
35+
with the ordering
36+
⟨i, v, _⟩ ≤ ⟨i', v', _⟩ ↔ i ⊆ i' ∧ ∀ a ∈ i, v a = v' a
37+
The hypothesis that `X` is normal implies that a maximal element must have `i = univ`.
38+
Point-finiteness of `u` (hypothesis `uf`) implies that
39+
the least upper bound of a chain in `T` again yields a covering of `s`.
40+
41+
Compare proofs in
42+
* https://ncatlab.org/nlab/show/shrinking+lemma#ShrinkingLemmaForLocallyFiniteCountableCovers
43+
* Bourbaki, General Topology, Chapter IX, §4.3
44+
* Dugundji, Topology, Chapter VII, Theorem 6.1
45+
-/

0 commit comments

Comments
 (0)