Gδ
sets
In this file we define Gδ
sets and prove their basic properties.
Main definitions
is_Gδ
: a sets
is aGδ
set if it can be represented as an intersection of countably many open sets;residual
: the filter of residual sets. A sets
is called residual if it includes a denseGδ
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
An open set is a Gδ set.
The union of two Gδ sets is a Gδ set.
The set of points where a function is continuous is a Gδ set.
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.