topology.G_delta

# Gδ sets

In this file we define Gδ sets and prove their basic properties.

## Main definitions

• is_Gδ: a set s is a Gδ set if it can be represented as an intersection of countably many open sets;

• residual: the filter of residual sets. A set s is called residual if it includes a dense Gδ set. In a Baire space (e.g., in a complete (e)metric space), residual sets form a filter.

For technical reasons, we define residual in any topological space but the definition agrees with the description above only in Baire spaces.

## Main results

We prove that finite or countable intersections of Gδ sets is a Gδ set. We also prove that the continuity set of a function from a topological space to an (e)metric space is a Gδ set.

## Tags

Gδ set, residual set

def is_Gδ {α : Type u_1} (s : set α) :
Prop

A Gδ set is a countable intersection of open sets.

Equations
theorem is_open.is_Gδ {α : Type u_1} {s : set α} (h : is_open s) :

An open set is a Gδ set.

theorem is_Gδ_univ {α : Type u_1}  :

theorem is_Gδ_bInter_of_open {α : Type u_1} {ι : Type u_4} {I : set ι} (hI : I.countable) {f : ι → set α} (hf : ∀ (i : ι), i Iis_open (f i)) :
is_Gδ (⋂ (i : ι) (H : i I), f i)

theorem is_Gδ_Inter_of_open {α : Type u_1} {ι : Type u_4} [encodable ι] {f : ι → set α} (hf : ∀ (i : ι), is_open (f i)) :
is_Gδ (⋂ (i : ι), f i)

theorem is_Gδ_sInter {α : Type u_1} {S : set (set α)} (h : ∀ (s : set α), s S) (hS : S.countable) :

A countable intersection of Gδ sets is a Gδ set.

theorem is_Gδ_Inter {α : Type u_1} {ι : Type u_4} [encodable ι] {s : ι → set α} (hs : ∀ (i : ι), is_Gδ (s i)) :
is_Gδ (⋂ (i : ι), s i)

theorem is_Gδ_bInter {α : Type u_1} {ι : Type u_4} {s : set ι} (hs : s.countable) {t : Π (i : ι), i sset α} (ht : ∀ (i : ι) (H : i s), is_Gδ (t i H)) :
is_Gδ (⋂ (i : ι) (H : i s), t i H)

theorem is_Gδ.inter {α : Type u_1} {s t : set α} (hs : is_Gδ s) (ht : is_Gδ t) :
is_Gδ (s t)

theorem is_Gδ.union {α : Type u_1} {s t : set α} (hs : is_Gδ s) (ht : is_Gδ t) :
is_Gδ (s t)

The union of two Gδ sets is a Gδ set.

theorem is_Gδ_set_of_continuous_at_of_countably_generated_uniformity {α : Type u_1} {β : Type u_2} (hU : (𝓤 β).is_countably_generated) (f : α → β) :
is_Gδ {x : α | x}

theorem is_Gδ_set_of_continuous_at {α : Type u_1} {β : Type u_2} (f : α → β) :
is_Gδ {x : α | x}

The set of points where a function is continuous is a Gδ set.

def residual (α : Type u_1)  :

A set s is called residual if it includes a dense Gδ set. If α is a Baire space (e.g., a complete metric space), then residual sets form a filter, see mem_residual.

For technical reasons we define the filter residual in any topological space but in a non-Baire space it is not useful because it may contain some non-residual sets.

Equations