# Documentation

Mathlib.Topology.ShrinkingLemma

# The shrinking lemma #

In this file we prove a few versions of the shrinking lemma. The lemma says that in a normal topological space a point finite open covering can be “shrunk”: for a point finite open covering u : ι → Set X there exists a refinement v : ι → Set X such that closure (v i) ⊆ u i.

For finite or countable coverings this lemma can be proved without the axiom of choice, see ncatlab for details. We only formalize the most general result that works for any covering but needs the axiom of choice.

We prove two versions of the lemma:

• exists_subset_iUnion_closure_subset deals with a covering of a closed set in a normal space;
• exists_iUnion_eq_closure_subset deals with a covering of the whole space.

## Tags #

normal space, shrinking lemma

theorem ShrinkingLemma.PartialRefinement.ext {ι : Type u_1} {X : Type u_2} :
∀ {inst : } {u : ιSet X} {s : Set X} (x y : ), x.toFun = y.toFunx.carrier = y.carrierx = y
theorem ShrinkingLemma.PartialRefinement.ext_iff {ι : Type u_1} {X : Type u_2} :
∀ {inst : } {u : ιSet X} {s : Set X} (x y : ), x = y x.toFun = y.toFun x.carrier = y.carrier
structure ShrinkingLemma.PartialRefinement {ι : Type u_1} {X : Type u_2} [] (u : ιSet X) (s : Set X) :
Type (max u_1 u_2)
• toFun : ιSet X

A family of sets that form a partial refinement of u.

• carrier : Set ι

The set of indexes i such that i-th set is already shrunk.

• isOpen : ∀ (i : ι),

Each set from the partially refined family is open.

• subset_iUnion : s✝ ⋃ (i : ι),

The partially refined family still covers the set.

• closure_subset : ∀ {i : ι}, i s.carrier

For each i ∈ carrier, the original set includes the closure of the refined set.

• apply_eq : ∀ {i : ι}, ¬i s.carrier

Sets that correspond to i ∉ carrier are not modified.

Auxiliary definition for the proof of the shrinking lemma. A partial refinement of a covering ⋃ i, u i of a set s is a map v : ι → Set X and a set carrier : Set ι such that

• s ⊆ ⋃ i, v i;
• all v i are open;
• if i ∈ carrier v, then closure (v i) ⊆ u i;
• if i ∉ carrier, then v i = u i.

This type is equipped with the following partial order: v ≤ v' if v.carrier ⊆ v'.carrier and v i = v' i for i ∈ v.carrier. We will use Zorn's lemma to prove that this type has a maximal element, then show that the maximal element must have carrier = univ.

Instances For
instance ShrinkingLemma.PartialRefinement.instCoeFunPartialRefinementForAllSet {ι : Type u_1} {X : Type u_2} [] {u : ιSet X} {s : Set X} :
CoeFun () fun x => ιSet X
theorem ShrinkingLemma.PartialRefinement.subset {ι : Type u_1} {X : Type u_2} [] {u : ιSet X} {s : Set X} (v : ) (i : ι) :
instance ShrinkingLemma.PartialRefinement.instPartialOrderPartialRefinement {ι : Type u_1} {X : Type u_2} [] {u : ιSet X} {s : Set X} :
theorem ShrinkingLemma.PartialRefinement.apply_eq_of_chain {ι : Type u_1} {X : Type u_2} [] {u : ιSet X} {s : Set X} {c : } (hc : IsChain (fun x x_1 => x x_1) c) {v₁ : } {v₂ : } (h₁ : v₁ c) (h₂ : v₂ c) {i : ι} (hi₁ : i v₁.carrier) (hi₂ : i v₂.carrier) :

If two partial refinements v₁, v₂ belong to a chain (hence, they are comparable) and i belongs to the carriers of both partial refinements, then v₁ i = v₂ i.

def ShrinkingLemma.PartialRefinement.chainSupCarrier {ι : Type u_1} {X : Type u_2} [] {u : ιSet X} {s : Set X} (c : ) :
Set ι

The carrier of the least upper bound of a non-empty chain of partial refinements is the union of their carriers.

Instances For
def ShrinkingLemma.PartialRefinement.find {ι : Type u_1} {X : Type u_2} [] {u : ιSet X} {s : Set X} (c : ) (ne : ) (i : ι) :

Choice of an element of a nonempty chain of partial refinements. If i belongs to one of carrier v, v ∈ c, then find c ne i is one of these partial refinements.

Instances For
theorem ShrinkingLemma.PartialRefinement.find_mem {ι : Type u_1} {X : Type u_2} [] {u : ιSet X} {s : Set X} {c : } (i : ι) (ne : ) :
theorem ShrinkingLemma.PartialRefinement.mem_find_carrier_iff {ι : Type u_1} {X : Type u_2} [] {u : ιSet X} {s : Set X} {c : } {i : ι} (ne : ) :
i ().carrier
theorem ShrinkingLemma.PartialRefinement.find_apply_of_mem {ι : Type u_1} {X : Type u_2} [] {u : ιSet X} {s : Set X} {c : } (hc : IsChain (fun x x_1 => x x_1) c) (ne : ) {i : ι} {v : } (hv : v c) (hi : i v.carrier) :
def ShrinkingLemma.PartialRefinement.chainSup {ι : Type u_1} {X : Type u_2} [] {u : ιSet X} {s : Set X} (c : ) (hc : IsChain (fun x x_1 => x x_1) c) (ne : ) (hfin : ∀ (x : X), x sSet.Finite {i | x u i}) (hU : s ⋃ (i : ι), u i) :

Least upper bound of a nonempty chain of partial refinements.

Instances For
theorem ShrinkingLemma.PartialRefinement.le_chainSup {ι : Type u_1} {X : Type u_2} [] {u : ιSet X} {s : Set X} {c : } (hc : IsChain (fun x x_1 => x x_1) c) (ne : ) (hfin : ∀ (x : X), x sSet.Finite {i | x u i}) (hU : s ⋃ (i : ι), u i) {v : } (hv : v c) :

chainSup hu c hc ne hfin hU is an upper bound of the chain c.

theorem ShrinkingLemma.PartialRefinement.exists_gt {ι : Type u_1} {X : Type u_2} [] [] {u : ιSet X} {s : Set X} (v : ) (hs : ) (i : ι) (hi : ¬i v.carrier) :
v', v < v'

If s is a closed set, v is a partial refinement, and i is an index such that i ∉ v.carrier, then there exists a partial refinement that is strictly greater than v.

theorem exists_subset_iUnion_closure_subset {ι : Type u_1} {X : Type u_2} [] [] {u : ιSet X} {s : Set X} (hs : ) (uo : ∀ (i : ι), IsOpen (u i)) (uf : ∀ (x : X), x sSet.Finite {i | x u i}) (us : s ⋃ (i : ι), u i) :
v, s (∀ (i : ι), IsOpen (v i)) ∀ (i : ι), closure (v i) u i

Shrinking lemma. A point-finite open cover of a closed subset of a normal space can be "shrunk" to a new open cover so that the closure of each new open set is contained in the corresponding original open set.

theorem exists_subset_iUnion_closed_subset {ι : Type u_1} {X : Type u_2} [] [] {u : ιSet X} {s : Set X} (hs : ) (uo : ∀ (i : ι), IsOpen (u i)) (uf : ∀ (x : X), x sSet.Finite {i | x u i}) (us : s ⋃ (i : ι), u i) :
v, s (∀ (i : ι), IsClosed (v i)) ∀ (i : ι), v i u i

Shrinking lemma. A point-finite open cover of a closed subset of a normal space can be "shrunk" to a new closed cover so that each new closed set is contained in the corresponding original open set. See also exists_subset_iUnion_closure_subset for a stronger statement.

theorem exists_iUnion_eq_closure_subset {ι : Type u_1} {X : Type u_2} [] [] {u : ιSet X} (uo : ∀ (i : ι), IsOpen (u i)) (uf : ∀ (x : X), Set.Finite {i | x u i}) (uU : ⋃ (i : ι), u i = Set.univ) :
v, = Set.univ (∀ (i : ι), IsOpen (v i)) ∀ (i : ι), closure (v i) u i

Shrinking lemma. A point-finite open cover of a closed subset of a normal space can be "shrunk" to a new open cover so that the closure of each new open set is contained in the corresponding original open set.

theorem exists_iUnion_eq_closed_subset {ι : Type u_1} {X : Type u_2} [] [] {u : ιSet X} (uo : ∀ (i : ι), IsOpen (u i)) (uf : ∀ (x : X), Set.Finite {i | x u i}) (uU : ⋃ (i : ι), u i = Set.univ) :
v, = Set.univ (∀ (i : ι), IsClosed (v i)) ∀ (i : ι), v i u i

Shrinking lemma. A point-finite open cover of a closed subset of a normal space can be "shrunk" to a new closed cover so that each of the new closed sets is contained in the corresponding original open set. See also exists_iUnion_eq_closure_subset for a stronger statement.