Continuous partition of unity

In this file we define PartitionOfUnity (ι X : Type*) [TopologicalSpace X] (s : Set X := univ) to be a continuous partition of unity on s indexed by ι. More precisely, f : PartitionOfUnity ι X s is a collection of continuous functions f i : C(X, ℝ), i : ι, such that

• the supports of f i form a locally finite family of sets;
• each f i is nonnegative;
• ∑ᶠ i, f i x = 1 for all x ∈ s;
• ∑ᶠ i, f i x ≤ 1 for all x : X.

In the case s = univ the last assumption follows from the previous one but it is convenient to have this assumption in the case s ≠ univ.

We also define a bump function covering, BumpCovering (ι X : Type*) [TopologicalSpace X] (s : Set X := univ), to be a collection of functions f i : C(X, ℝ), i : ι, such that

• the supports of f i form a locally finite family of sets;
• each f i is nonnegative;
• for each x ∈ s there exists i : ι such that f i y = 1 in a neighborhood of x.

The term is motivated by the smooth case.

If f is a bump function covering indexed by a linearly ordered type, then g i x = f i x * ∏ᶠ j < i, (1 - f j x) is a partition of unity, see BumpCovering.toPartitionOfUnity. Note that only finitely many terms 1 - f j x are not equal to one, so this product is well-defined.

Note that g i x = ∏ᶠ j ≤ i, (1 - f j x) - ∏ᶠ j < i, (1 - f j x), so most terms in the sum ∑ᶠ i, g i x cancel, and we get ∑ᶠ i, g i x = 1 - ∏ᶠ i, (1 - f i x), and the latter product equals zero because one of f i x is equal to one.

We say that a partition of unity or a bump function covering f is subordinate to a family of sets U i, i : ι, if the closure of the support of each f i is included in U i. We use Urysohn's Lemma to prove that a locally finite open covering of a normal topological space admits a subordinate bump function covering (hence, a subordinate partition of unity), see BumpCovering.exists_isSubordinate_of_locallyFinite. If X is a paracompact space, then any open covering admits a locally finite refinement, hence it admits a subordinate bump function covering and a subordinate partition of unity, see BumpCovering.exists_isSubordinate.

We also provide two slightly more general versions of these lemmas, BumpCovering.exists_isSubordinate_of_locallyFinite_of_prop and BumpCovering.exists_isSubordinate_of_prop, to be used later in the construction of a smooth partition of unity.

Implementation notes

Most (if not all) books only define a partition of unity of the whole space. However, quite a few proofs only deal with f i such that tsupport (f i) meets a specific closed subset, and it is easier to formalize these proofs if we don't have other functions right away.

We use WellOrderingRel j i instead of j < i in the definition of BumpCovering.toPartitionOfUnity to avoid a [LinearOrder ι] assumption. While WellOrderingRel j i is a well order, not only a strict linear order, we never use this property.

Tags

partition of unity, bump function, Urysohn's lemma, normal space, paracompact space

structure PartitionOfUnity (ι : Type u_1) (X : Type u_2) [TopologicalSpace X] (s : Set X := Set.univ) : Type (max u_1 u_2)

A continuous partition of unity on a set s : Set X is a collection of continuous functions f i such that

• the supports of f i form a locally finite family of sets, i.e., for every point x : X there exists a neighborhood U ∋ x such that all but finitely many functions f i are zero on U;
• the functions f i are nonnegative;
• the sum ∑ᶠ i, f i x is equal to one for every x ∈ s and is less than or equal to one otherwise.

If X is a normal paracompact space, then PartitionOfUnity.exists_isSubordinate guarantees that for every open covering U : Set (Set X) of s there exists a partition of unity that is subordinate to U.

• toFun : ι → C(X, ℝ)

  The collection of continuous functions underlying this partition of unity

• locallyFinite' : LocallyFinite fun (i : ι) => Function.support ⇑(self.toFun i)

  the supports of the underlying functions are a locally finite family of sets

• nonneg' : 0 ≤ self.toFun

  the functions are non-negative

• sum_eq_one' : ∀ x ∈ s, ∑ᶠ (i : ι), (self.toFun i) x = 1

  the functions sum up to one on s

• sum_le_one' : ∀ (x : X), ∑ᶠ (i : ι), (self.toFun i) x ≤ 1

  the functions sum up to at most one, globally

structure BumpCovering (ι : Type u_1) (X : Type u_2) [TopologicalSpace X] (s : Set X := Set.univ) : Type (max u_1 u_2)

A BumpCovering ι X s is an indexed family of functions f i, i : ι, such that

• the supports of f i form a locally finite family of sets, i.e., for every point x : X there exists a neighborhood U ∋ x such that all but finitely many functions f i are zero on U;
• for all i, x we have 0 ≤ f i x ≤ 1;
• each point x ∈ s belongs to the interior of {x | f i x = 1} for some i.

One of the main use cases for a BumpCovering is to define a PartitionOfUnity, see BumpCovering.toPartitionOfUnity, but some proofs can directly use a BumpCovering instead of a PartitionOfUnity.

If X is a normal paracompact space, then BumpCovering.exists_isSubordinate guarantees that for every open covering U : Set (Set X) of s there exists a BumpCovering of s that is subordinate to U.

• toFun : ι → C(X, ℝ)

  The collections of continuous functions underlying this bump covering

• locallyFinite' : LocallyFinite fun (i : ι) => Function.support ⇑(self.toFun i)

  the supports of the underlying functions are a locally finite family of sets

• nonneg' : 0 ≤ self.toFun

  the functions are non-negative

• le_one' : self.toFun ≤ 1

  the functions are each at most one

• eventuallyEq_one' : ∀ x ∈ s, ∃ (i : ι), ⇑(self.toFun i) =ᶠ[nhds x] 1

  Each point x ∈ s belongs to the interior of {x | f i x = 1} for some i.

instance PartitionOfUnity.instFunLikeContinuousMapReal {ι : Type u} {X : Type v} [TopologicalSpace X] {s : Set X} : FunLike (PartitionOfUnity ι X s) ι C(X, ℝ)

Equations

• PartitionOfUnity.instFunLikeContinuousMapReal = { coe := PartitionOfUnity.toFun, coe_injective' := ⋯ }

theorem PartitionOfUnity.locallyFinite {ι : Type u} {X : Type v} [TopologicalSpace X] {s : Set X} (f : PartitionOfUnity ι X s) : LocallyFinite fun (i : ι) => Function.support ⇑(f i)

theorem PartitionOfUnity.locallyFinite_tsupport {ι : Type u} {X : Type v} [TopologicalSpace X] {s : Set X} (f : PartitionOfUnity ι X s) : LocallyFinite fun (i : ι) => tsupport ⇑(f i)

theorem PartitionOfUnity.nonneg {ι : Type u} {X : Type v} [TopologicalSpace X] {s : Set X} (f : PartitionOfUnity ι X s) (i : ι) (x : X) : 0 ≤ (f i) x

theorem PartitionOfUnity.sum_eq_one {ι : Type u} {X : Type v} [TopologicalSpace X] {s : Set X} (f : PartitionOfUnity ι X s) {x : X} (hx : x ∈ s) : ∑ᶠ (i : ι), (f i) x = 1

theorem PartitionOfUnity.exists_pos {ι : Type u} {X : Type v} [TopologicalSpace X] {s : Set X} (f : PartitionOfUnity ι X s) {x : X} (hx : x ∈ s) : ∃ (i : ι), 0 < (f i) x

If f is a partition of unity on s, then for every x ∈ s there exists an index i such that 0 < f i x.

theorem PartitionOfUnity.sum_le_one {ι : Type u} {X : Type v} [TopologicalSpace X] {s : Set X} (f : PartitionOfUnity ι X s) (x : X) : ∑ᶠ (i : ι), (f i) x ≤ 1

theorem PartitionOfUnity.sum_nonneg {ι : Type u} {X : Type v} [TopologicalSpace X] {s : Set X} (f : PartitionOfUnity ι X s) (x : X) : 0 ≤ ∑ᶠ (i : ι), (f i) x

theorem PartitionOfUnity.le_one {ι : Type u} {X : Type v} [TopologicalSpace X] {s : Set X} (f : PartitionOfUnity ι X s) (i : ι) (x : X) : (f i) x ≤ 1

def PartitionOfUnity.finsupport {ι : Type u} {X : Type v} [TopologicalSpace X] {s : Set X} (ρ : PartitionOfUnity ι X s) (x₀ : X) : Finset ι

The support of a partition of unity at a point x₀ as a Finset. This is the set of i : ι such that x₀ ∈ support f i, i.e. f i ≠ x₀.

Equations

• ρ.finsupport x₀ = ⋯.toFinset

@[simp]

theorem PartitionOfUnity.mem_finsupport {ι : Type u} {X : Type v} [TopologicalSpace X] {s : Set X} (ρ : PartitionOfUnity ι X s) (x₀ : X) {i : ι} : i ∈ ρ.finsupport x₀ ↔ i ∈ Function.support fun (i : ι) => (ρ i) x₀

@[simp]

theorem PartitionOfUnity.coe_finsupport {ι : Type u} {X : Type v} [TopologicalSpace X] {s : Set X} (ρ : PartitionOfUnity ι X s) (x₀ : X) : ↑(ρ.finsupport x₀) = Function.support fun (i : ι) => (ρ i) x₀

theorem PartitionOfUnity.sum_finsupport {ι : Type u} {X : Type v} [TopologicalSpace X] {s : Set X} (ρ : PartitionOfUnity ι X s) {x₀ : X} (hx₀ : x₀ ∈ s) : ∑ i ∈ ρ.finsupport x₀, (ρ i) x₀ = 1

theorem PartitionOfUnity.sum_finsupport' {ι : Type u} {X : Type v} [TopologicalSpace X] {s : Set X} (ρ : PartitionOfUnity ι X s) {x₀ : X} (hx₀ : x₀ ∈ s) {I : Finset ι} (hI : ρ.finsupport x₀ ⊆ I) : ∑ i ∈ I, (ρ i) x₀ = 1

theorem PartitionOfUnity.sum_finsupport_smul_eq_finsum {ι : Type u} {X : Type v} [TopologicalSpace X] {s : Set X} (ρ : PartitionOfUnity ι X s) {x₀ : X} {M : Type u_2} [AddCommGroup M] [Module ℝ M] (φ : ι → X → M) : ∑ i ∈ ρ.finsupport x₀, (ρ i) x₀ • φ i x₀ = ∑ᶠ (i : ι), (ρ i) x₀ • φ i x₀

theorem PartitionOfUnity.finite_tsupport {ι : Type u} {X : Type v} [TopologicalSpace X] {s : Set X} (ρ : PartitionOfUnity ι X s) (x₀ : X) : {i : ι | x₀ ∈ tsupport ⇑(ρ i)}.Finite

The tsupports of a partition of unity are locally finite.

def PartitionOfUnity.fintsupport {ι : Type u} {X : Type v} [TopologicalSpace X] {s : Set X} (ρ : PartitionOfUnity ι X s) (x₀ : X) : Finset ι

The tsupport of a partition of unity at a point x₀ as a Finset. This is the set of i : ι such that x₀ ∈ tsupport f i.

Equations

• ρ.fintsupport x₀ = ⋯.toFinset

theorem PartitionOfUnity.mem_fintsupport_iff {ι : Type u} {X : Type v} [TopologicalSpace X] {s : Set X} (ρ : PartitionOfUnity ι X s) (x₀ : X) (i : ι) : i ∈ ρ.fintsupport x₀ ↔ x₀ ∈ tsupport ⇑(ρ i)

theorem PartitionOfUnity.eventually_fintsupport_subset {ι : Type u} {X : Type v} [TopologicalSpace X] {s : Set X} (ρ : PartitionOfUnity ι X s) (x₀ : X) : ∀ᶠ (y : X) in nhds x₀, ρ.fintsupport y ⊆ ρ.fintsupport x₀

theorem PartitionOfUnity.finsupport_subset_fintsupport {ι : Type u} {X : Type v} [TopologicalSpace X] {s : Set X} (ρ : PartitionOfUnity ι X s) (x₀ : X) : ρ.finsupport x₀ ⊆ ρ.fintsupport x₀

theorem PartitionOfUnity.eventually_finsupport_subset {ι : Type u} {X : Type v} [TopologicalSpace X] {s : Set X} (ρ : PartitionOfUnity ι X s) (x₀ : X) : ∀ᶠ (y : X) in nhds x₀, ρ.finsupport y ⊆ ρ.fintsupport x₀

theorem PartitionOfUnity.continuous_smul {ι : Type u} {X : Type v} [TopologicalSpace X] {E : Type u_1} [AddCommMonoid E] [SMulWithZero ℝ E] [TopologicalSpace E] [ContinuousSMul ℝ E] {s : Set X} (f : PartitionOfUnity ι X s) {g : X → E} {i : ι} (hg : ∀ x ∈ tsupport ⇑(f i), ContinuousAt g x) : Continuous fun (x : X) => (f i) x • g x

If f is a partition of unity on s : Set X and g : X → E is continuous at every point of the topological support of some f i, then fun x ↦ f i x • g x is continuous on the whole space.

theorem PartitionOfUnity.continuous_finsum_smul {ι : Type u} {X : Type v} [TopologicalSpace X] {E : Type u_1} [AddCommMonoid E] [SMulWithZero ℝ E] [TopologicalSpace E] [ContinuousSMul ℝ E] {s : Set X} (f : PartitionOfUnity ι X s) [ContinuousAdd E] {g : ι → X → E} (hg : ∀ (i : ι), ∀ x ∈ tsupport ⇑(f i), ContinuousAt (g i) x) : Continuous fun (x : X) => ∑ᶠ (i : ι), (f i) x • g i x

If f is a partition of unity on a set s : Set X and g : ι → X → E is a family of functions such that each g i is continuous at every point of the topological support of f i, then the sum fun x ↦ ∑ᶠ i, f i x • g i x is continuous on the whole space.

def PartitionOfUnity.IsSubordinate {ι : Type u} {X : Type v} [TopologicalSpace X] {s : Set X} (f : PartitionOfUnity ι X s) (U : ι → Set X) : Prop

A partition of unity f i is subordinate to a family of sets U i indexed by the same type if for each i the closure of the support of f i is a subset of U i.

Equations

• f.IsSubordinate U = ∀ (i : ι), tsupport ⇑(f i) ⊆ U i

theorem PartitionOfUnity.exists_finset_nhd' {ι : Type u} {X : Type v} [TopologicalSpace X] {s : Set X} (ρ : PartitionOfUnity ι X s) (x₀ : X) : ∃ (I : Finset ι), (∀ᶠ (x : X) in nhdsWithin x₀ s, ∑ i ∈ I, (ρ i) x = 1) ∧ ∀ᶠ (x : X) in nhds x₀, (Function.support fun (x_1 : ι) => (ρ x_1) x) ⊆ ↑I

theorem PartitionOfUnity.exists_finset_nhd {ι : Type u} {X : Type v} [TopologicalSpace X] (ρ : PartitionOfUnity ι X) (x₀ : X) : ∃ (I : Finset ι), ∀ᶠ (x : X) in nhds x₀, ∑ i ∈ I, (ρ i) x = 1 ∧ (Function.support fun (x_1 : ι) => (ρ x_1) x) ⊆ ↑I

theorem PartitionOfUnity.exists_finset_nhd_support_subset {ι : Type u} {X : Type v} [TopologicalSpace X] {s : Set X} {f : PartitionOfUnity ι X s} {U : ι → Set X} (hso : f.IsSubordinate U) (ho : ∀ (i : ι), IsOpen (U i)) (x : X) : ∃ (is : Finset ι), ∃ n ∈ nhds x, n ⊆ ⋂ i ∈ is, U i ∧ ∀ z ∈ n, (Function.support fun (x : ι) => (f x) z) ⊆ ↑is

theorem PartitionOfUnity.IsSubordinate.continuous_finsum_smul {ι : Type u} {X : Type v} [TopologicalSpace X] {E : Type u_1} [AddCommMonoid E] [SMulWithZero ℝ E] [TopologicalSpace E] [ContinuousSMul ℝ E] {s : Set X} {f : PartitionOfUnity ι X s} [ContinuousAdd E] {U : ι → Set X} (ho : ∀ (i : ι), IsOpen (U i)) (hf : f.IsSubordinate U) {g : ι → X → E} (hg : ∀ (i : ι), ContinuousOn (g i) (U i)) : Continuous fun (x : X) => ∑ᶠ (i : ι), (f i) x • g i x

If f is a partition of unity that is subordinate to a family of open sets U i and g : ι → X → E is a family of functions such that each g i is continuous on U i, then the sum fun x ↦ ∑ᶠ i, f i x • g i x is a continuous function.

instance BumpCovering.instFunLikeContinuousMapReal {ι : Type u} {X : Type v} [TopologicalSpace X] {s : Set X} : FunLike (BumpCovering ι X s) ι C(X, ℝ)

Equations

• BumpCovering.instFunLikeContinuousMapReal = { coe := BumpCovering.toFun, coe_injective' := ⋯ }

@[simp]

theorem BumpCovering.toFun_eq_coe {ι : Type u} {X : Type v} [TopologicalSpace X] {s : Set X} (f : BumpCovering ι X s) : f.toFun = ⇑f

theorem BumpCovering.locallyFinite {ι : Type u} {X : Type v} [TopologicalSpace X] {s : Set X} (f : BumpCovering ι X s) : LocallyFinite fun (i : ι) => Function.support ⇑(f i)

theorem BumpCovering.locallyFinite_tsupport {ι : Type u} {X : Type v} [TopologicalSpace X] {s : Set X} (f : BumpCovering ι X s) : LocallyFinite fun (i : ι) => tsupport ⇑(f i)

theorem BumpCovering.point_finite {ι : Type u} {X : Type v} [TopologicalSpace X] {s : Set X} (f : BumpCovering ι X s) (x : X) : {i : ι | (f i) x ≠ 0}.Finite

theorem BumpCovering.nonneg {ι : Type u} {X : Type v} [TopologicalSpace X] {s : Set X} (f : BumpCovering ι X s) (i : ι) (x : X) : 0 ≤ (f i) x

theorem BumpCovering.le_one {ι : Type u} {X : Type v} [TopologicalSpace X] {s : Set X} (f : BumpCovering ι X s) (i : ι) (x : X) : (f i) x ≤ 1

def BumpCovering.single {ι : Type u} {X : Type v} [TopologicalSpace X] (i : ι) (s : Set X) : BumpCovering ι X s

A BumpCovering that consists of a single function, uniformly equal to one, defined as an example for Inhabited instance.

Equations

• BumpCovering.single i s = { toFun := Pi.single i 1, locallyFinite' := ⋯, nonneg' := ⋯, le_one' := ⋯, eventuallyEq_one' := ⋯ }

@[simp]

theorem BumpCovering.coe_single {ι : Type u} {X : Type v} [TopologicalSpace X] (i : ι) (s : Set X) : ⇑(BumpCovering.single i s) = Pi.single i 1

instance BumpCovering.instInhabited {ι : Type u} {X : Type v} [TopologicalSpace X] {s : Set X} [Inhabited ι] : Inhabited (BumpCovering ι X s)

Equations

• BumpCovering.instInhabited = { default := BumpCovering.single default s }

def BumpCovering.IsSubordinate {ι : Type u} {X : Type v} [TopologicalSpace X] {s : Set X} (f : BumpCovering ι X s) (U : ι → Set X) : Prop

A collection of bump functions f i is subordinate to a family of sets U i indexed by the same type if for each i the closure of the support of f i is a subset of U i.

Equations

• f.IsSubordinate U = ∀ (i : ι), tsupport ⇑(f i) ⊆ U i

theorem BumpCovering.IsSubordinate.mono {ι : Type u} {X : Type v} [TopologicalSpace X] {s : Set X} {f : BumpCovering ι X s} {U V : ι → Set X} (hU : f.IsSubordinate U) (hV : ∀ (i : ι), U i ⊆ V i) : f.IsSubordinate V

theorem BumpCovering.exists_isSubordinate_of_locallyFinite_of_prop {ι : Type u} {X : Type v} [TopologicalSpace X] {s : Set X} [NormalSpace X] (p : (X → ℝ) → Prop) (h01 : ∀ (s t : Set X), IsClosed s → IsClosed t → Disjoint s t → ∃ (f : C(X, ℝ)), p ⇑f ∧ Set.EqOn (⇑f) 0 s ∧ Set.EqOn (⇑f) 1 t ∧ ∀ (x : X), f x ∈ Set.Icc 0 1) (hs : IsClosed s) (U : ι → Set X) (ho : ∀ (i : ι), IsOpen (U i)) (hf : LocallyFinite U) (hU : s ⊆ ⋃ (i : ι), U i) : ∃ (f : BumpCovering ι X s), (∀ (i : ι), p ⇑(f i)) ∧ f.IsSubordinate U

If X is a normal topological space and U i, i : ι, is a locally finite open covering of a closed set s, then there exists a BumpCovering ι X s that is subordinate to U. If X is a paracompact space, then the assumption hf : LocallyFinite U can be omitted, see BumpCovering.exists_isSubordinate. This version assumes that p : (X → ℝ) → Prop is a predicate that satisfies Urysohn's lemma, and provides a BumpCovering such that each function of the covering satisfies p.

theorem BumpCovering.exists_isSubordinate_of_locallyFinite {ι : Type u} {X : Type v} [TopologicalSpace X] {s : Set X} [NormalSpace X] (hs : IsClosed s) (U : ι → Set X) (ho : ∀ (i : ι), IsOpen (U i)) (hf : LocallyFinite U) (hU : s ⊆ ⋃ (i : ι), U i) : ∃ (f : BumpCovering ι X s), f.IsSubordinate U

If X is a normal topological space and U i, i : ι, is a locally finite open covering of a closed set s, then there exists a BumpCovering ι X s that is subordinate to U. If X is a paracompact space, then the assumption hf : LocallyFinite U can be omitted, see BumpCovering.exists_isSubordinate.

theorem BumpCovering.exists_isSubordinate_of_prop {ι : Type u} {X : Type v} [TopologicalSpace X] {s : Set X} [NormalSpace X] [ParacompactSpace X] (p : (X → ℝ) → Prop) (h01 : ∀ (s t : Set X), IsClosed s → IsClosed t → Disjoint s t → ∃ (f : C(X, ℝ)), p ⇑f ∧ Set.EqOn (⇑f) 0 s ∧ Set.EqOn (⇑f) 1 t ∧ ∀ (x : X), f x ∈ Set.Icc 0 1) (hs : IsClosed s) (U : ι → Set X) (ho : ∀ (i : ι), IsOpen (U i)) (hU : s ⊆ ⋃ (i : ι), U i) : ∃ (f : BumpCovering ι X s), (∀ (i : ι), p ⇑(f i)) ∧ f.IsSubordinate U

If X is a paracompact normal topological space and U is an open covering of a closed set s, then there exists a BumpCovering ι X s that is subordinate to U. This version assumes that p : (X → ℝ) → Prop is a predicate that satisfies Urysohn's lemma, and provides a BumpCovering such that each function of the covering satisfies p.

theorem BumpCovering.exists_isSubordinate {ι : Type u} {X : Type v} [TopologicalSpace X] {s : Set X} [NormalSpace X] [ParacompactSpace X] (hs : IsClosed s) (U : ι → Set X) (ho : ∀ (i : ι), IsOpen (U i)) (hU : s ⊆ ⋃ (i : ι), U i) : ∃ (f : BumpCovering ι X s), f.IsSubordinate U

If X is a paracompact normal topological space and U is an open covering of a closed set s, then there exists a BumpCovering ι X s that is subordinate to U.

theorem BumpCovering.exists_isSubordinate_of_locallyFinite_of_prop_t2space {ι : Type u} {X : Type v} [TopologicalSpace X] {s : Set X} [LocallyCompactSpace X] [T2Space X] (p : (X → ℝ) → Prop) (h01 : ∀ (s t : Set X), IsClosed s → IsCompact t → Disjoint s t → ∃ (f : C(X, ℝ)), p ⇑f ∧ Set.EqOn (⇑f) 0 s ∧ Set.EqOn (⇑f) 1 t ∧ ∀ (x : X), f x ∈ Set.Icc 0 1) (hs : IsCompact s) (U : ι → Set X) (ho : ∀ (i : ι), IsOpen (U i)) (hf : LocallyFinite U) (hU : s ⊆ ⋃ (i : ι), U i) : ∃ (f : BumpCovering ι X s), (∀ (i : ι), p ⇑(f i)) ∧ f.IsSubordinate U ∧ ∀ (i : ι), HasCompactSupport ⇑(f i)

If X is a locally compact T2 topological space and U i, i : ι, is a locally finite open covering of a compact set s, then there exists a BumpCovering ι X s that is subordinate to U. If X is a paracompact space, then the assumption hf : LocallyFinite U can be omitted, see BumpCovering.exists_isSubordinate. This version assumes that p : (X → ℝ) → Prop is a predicate that satisfies Urysohn's lemma, and provides a BumpCovering such that each function of the covering satisfies p.

theorem BumpCovering.exists_isSubordinate_hasCompactSupport_of_locallyFinite_t2space {ι : Type u} {X : Type v} [TopologicalSpace X] {s : Set X} [LocallyCompactSpace X] [T2Space X] (hs : IsCompact s) (U : ι → Set X) (ho : ∀ (i : ι), IsOpen (U i)) (hf : LocallyFinite U) (hU : s ⊆ ⋃ (i : ι), U i) : ∃ (f : BumpCovering ι X s), f.IsSubordinate U ∧ ∀ (i : ι), HasCompactSupport ⇑(f i)

If X is a normal topological space and U i, i : ι, is a locally finite open covering of a closed set s, then there exists a BumpCovering ι X s that is subordinate to U. If X is a paracompact space, then the assumption hf : LocallyFinite U can be omitted, see BumpCovering.exists_isSubordinate.

def BumpCovering.ind {ι : Type u} {X : Type v} [TopologicalSpace X] {s : Set X} (f : BumpCovering ι X s) (x : X) (hx : x ∈ s) : ι

Index of a bump function such that fs i =ᶠ[𝓝 x] 1.

Equations

• f.ind x hx = ⋯.choose

theorem BumpCovering.eventuallyEq_one {ι : Type u} {X : Type v} [TopologicalSpace X] {s : Set X} (f : BumpCovering ι X s) (x : X) (hx : x ∈ s) : ⇑(f (f.ind x hx)) =ᶠ[nhds x] 1

theorem BumpCovering.ind_apply {ι : Type u} {X : Type v} [TopologicalSpace X] {s : Set X} (f : BumpCovering ι X s) (x : X) (hx : x ∈ s) : (f (f.ind x hx)) x = 1

def BumpCovering.toPOUFun {ι : Type u} {X : Type v} [TopologicalSpace X] {s : Set X} (f : BumpCovering ι X s) (i : ι) (x : X) : ℝ

Partition of unity defined by a BumpCovering. We use this auxiliary definition to prove some properties of the new family of functions before bundling it into a PartitionOfUnity. Do not use this definition, use BumpCovering.toPartitionOfUnity instead.

The partition of unity is given by the formula g i x = f i x * ∏ᶠ j < i, (1 - f j x). In other words, g i x = ∏ᶠ j < i, (1 - f j x) - ∏ᶠ j ≤ i, (1 - f j x), so ∑ᶠ i, g i x = 1 - ∏ᶠ j, (1 - f j x). If x ∈ s, then one of f j x equals one, hence the product of 1 - f j x vanishes, and ∑ᶠ i, g i x = 1.

In order to avoid an assumption LinearOrder ι, we use WellOrderingRel instead of (<).

Equations

• f.toPOUFun i x = (f i) x * ∏ᶠ (j : ι) (_ : WellOrderingRel j i), (1 - (f j) x)

theorem BumpCovering.toPOUFun_zero_of_zero {ι : Type u} {X : Type v} [TopologicalSpace X] {s : Set X} (f : BumpCovering ι X s) {i : ι} {x : X} (h : (f i) x = 0) : f.toPOUFun i x = 0

theorem BumpCovering.support_toPOUFun_subset {ι : Type u} {X : Type v} [TopologicalSpace X] {s : Set X} (f : BumpCovering ι X s) (i : ι) : Function.support (f.toPOUFun i) ⊆ Function.support ⇑(f i)

theorem BumpCovering.toPOUFun_eq_mul_prod {ι : Type u} {X : Type v} [TopologicalSpace X] {s : Set X} (f : BumpCovering ι X s) (i : ι) (x : X) (t : Finset ι) (ht : ∀ (j : ι), WellOrderingRel j i → (f j) x ≠ 0 → j ∈ t) : f.toPOUFun i x = (f i) x * ∏ j ∈ Finset.filter (fun (j : ι) => WellOrderingRel j i) t, (1 - (f j) x)

theorem BumpCovering.sum_toPOUFun_eq {ι : Type u} {X : Type v} [TopologicalSpace X] {s : Set X} (f : BumpCovering ι X s) (x : X) : ∑ᶠ (i : ι), f.toPOUFun i x = 1 - ∏ᶠ (i : ι), (1 - (f i) x)

theorem BumpCovering.exists_finset_toPOUFun_eventuallyEq {ι : Type u} {X : Type v} [TopologicalSpace X] {s : Set X} (f : BumpCovering ι X s) (i : ι) (x : X) : ∃ (t : Finset ι), f.toPOUFun i =ᶠ[nhds x] ⇑(f i) * ⇑(∏ j ∈ Finset.filter (fun (j : ι) => WellOrderingRel j i) t, (1 - f j))

theorem BumpCovering.continuous_toPOUFun {ι : Type u} {X : Type v} [TopologicalSpace X] {s : Set X} (f : BumpCovering ι X s) (i : ι) : Continuous (f.toPOUFun i)

def BumpCovering.toPartitionOfUnity {ι : Type u} {X : Type v} [TopologicalSpace X] {s : Set X} (f : BumpCovering ι X s) : PartitionOfUnity ι X s

The partition of unity defined by a BumpCovering.

The partition of unity is given by the formula g i x = f i x * ∏ᶠ j < i, (1 - f j x). In other words, g i x = ∏ᶠ j < i, (1 - f j x) - ∏ᶠ j ≤ i, (1 - f j x), so ∑ᶠ i, g i x = 1 - ∏ᶠ j, (1 - f j x). If x ∈ s, then one of f j x equals one, hence the product of 1 - f j x vanishes, and ∑ᶠ i, g i x = 1.

In order to avoid an assumption LinearOrder ι, we use WellOrderingRel instead of (<).

Equations

• f.toPartitionOfUnity = { toFun := fun (i : ι) => { toFun := f.toPOUFun i, continuous_toFun := ⋯ }, locallyFinite' := ⋯, nonneg' := ⋯, sum_eq_one' := ⋯, sum_le_one' := ⋯ }

theorem BumpCovering.toPartitionOfUnity_apply {ι : Type u} {X : Type v} [TopologicalSpace X] {s : Set X} (f : BumpCovering ι X s) (i : ι) (x : X) : (f.toPartitionOfUnity i) x = (f i) x * ∏ᶠ (j : ι) (_ : WellOrderingRel j i), (1 - (f j) x)

theorem BumpCovering.toPartitionOfUnity_eq_mul_prod {ι : Type u} {X : Type v} [TopologicalSpace X] {s : Set X} (f : BumpCovering ι X s) (i : ι) (x : X) (t : Finset ι) (ht : ∀ (j : ι), WellOrderingRel j i → (f j) x ≠ 0 → j ∈ t) : (f.toPartitionOfUnity i) x = (f i) x * ∏ j ∈ Finset.filter (fun (j : ι) => WellOrderingRel j i) t, (1 - (f j) x)

theorem BumpCovering.exists_finset_toPartitionOfUnity_eventuallyEq {ι : Type u} {X : Type v} [TopologicalSpace X] {s : Set X} (f : BumpCovering ι X s) (i : ι) (x : X) : ∃ (t : Finset ι), ⇑(f.toPartitionOfUnity i) =ᶠ[nhds x] ⇑(f i) * ⇑(∏ j ∈ Finset.filter (fun (j : ι) => WellOrderingRel j i) t, (1 - f j))

theorem BumpCovering.toPartitionOfUnity_zero_of_zero {ι : Type u} {X : Type v} [TopologicalSpace X] {s : Set X} (f : BumpCovering ι X s) {i : ι} {x : X} (h : (f i) x = 0) : (f.toPartitionOfUnity i) x = 0

theorem BumpCovering.support_toPartitionOfUnity_subset {ι : Type u} {X : Type v} [TopologicalSpace X] {s : Set X} (f : BumpCovering ι X s) (i : ι) : Function.support ⇑(f.toPartitionOfUnity i) ⊆ Function.support ⇑(f i)

theorem BumpCovering.sum_toPartitionOfUnity_eq {ι : Type u} {X : Type v} [TopologicalSpace X] {s : Set X} (f : BumpCovering ι X s) (x : X) : ∑ᶠ (i : ι), (f.toPartitionOfUnity i) x = 1 - ∏ᶠ (i : ι), (1 - (f i) x)

theorem BumpCovering.IsSubordinate.toPartitionOfUnity {ι : Type u} {X : Type v} [TopologicalSpace X] {s : Set X} {f : BumpCovering ι X s} {U : ι → Set X} (h : f.IsSubordinate U) : f.toPartitionOfUnity.IsSubordinate U

instance PartitionOfUnity.instInhabited {ι : Type u} {X : Type v} [TopologicalSpace X] {s : Set X} [Inhabited ι] : Inhabited (PartitionOfUnity ι X s)

Equations

• PartitionOfUnity.instInhabited = { default := default.toPartitionOfUnity }

theorem PartitionOfUnity.exists_isSubordinate_of_locallyFinite {ι : Type u} {X : Type v} [TopologicalSpace X] {s : Set X} [NormalSpace X] (hs : IsClosed s) (U : ι → Set X) (ho : ∀ (i : ι), IsOpen (U i)) (hf : LocallyFinite U) (hU : s ⊆ ⋃ (i : ι), U i) : ∃ (f : PartitionOfUnity ι X s), f.IsSubordinate U

If X is a normal topological space and U is a locally finite open covering of a closed set s, then there exists a PartitionOfUnity ι X s that is subordinate to U. If X is a paracompact space, then the assumption hf : LocallyFinite U can be omitted, see BumpCovering.exists_isSubordinate.

theorem PartitionOfUnity.exists_isSubordinate {ι : Type u} {X : Type v} [TopologicalSpace X] {s : Set X} [NormalSpace X] [ParacompactSpace X] (hs : IsClosed s) (U : ι → Set X) (ho : ∀ (i : ι), IsOpen (U i)) (hU : s ⊆ ⋃ (i : ι), U i) : ∃ (f : PartitionOfUnity ι X s), f.IsSubordinate U

If X is a paracompact normal topological space and U is an open covering of a closed set s, then there exists a PartitionOfUnity ι X s that is subordinate to U.

theorem PartitionOfUnity.exists_isSubordinate_of_locallyFinite_t2space {ι : Type u} {X : Type v} [TopologicalSpace X] {s : Set X} [LocallyCompactSpace X] [T2Space X] (hs : IsCompact s) (U : ι → Set X) (ho : ∀ (i : ι), IsOpen (U i)) (hf : LocallyFinite U) (hU : s ⊆ ⋃ (i : ι), U i) : ∃ (f : PartitionOfUnity ι X s), f.IsSubordinate U ∧ ∀ (i : ι), HasCompactSupport ⇑(f i)

If X is a locally compact T2 topological space and U is a locally finite open covering of a compact set s, then there exists a PartitionOfUnity ι X s that is subordinate to U.

theorem exists_continuous_sum_one_of_isOpen_isCompact {X : Type v} [TopologicalSpace X] [T2Space X] [LocallyCompactSpace X] {n : ℕ} {t : Set X} {s : Fin n → Set X} (hs : ∀ (i : Fin n), IsOpen (s i)) (htcp : IsCompact t) (hst : t ⊆ ⋃ (i : Fin n), s i) : ∃ (f : Fin n → C(X, ℝ)), (∀ (i : Fin n), tsupport ⇑(f i) ⊆ s i) ∧ Set.EqOn (∑ i : Fin n, ⇑(f i)) 1 t ∧ (∀ (i : Fin n) (x : X), (f i) x ∈ Set.Icc 0 1) ∧ ∀ (i : Fin n), HasCompactSupport ⇑(f i)

A variation of Urysohn's lemma.

In a locally compact T2 space X, for a compact set t and a finite family of open sets {s i}_i such that t ⊆ ⋃ i, s i, there is a family of compactly supported continuous functions {f i}_i supported in s i, ∑ i, f i x = 1 on t and 0 ≤ f i x ≤ 1.