Gδ sets #
In this file we define
Gδ sets and prove their basic properties.
Main definitions #
is_Gδ: a set
Gδset if it can be represented as an intersection of countably many open sets;
residual: the filter of residual sets. A set
sis called residual if it includes a dense
Gδset. In a Baire space (e.g., in a complete (e)metric space), residual sets form a filter.
For technical reasons, we define
residualin 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.
Gδ set, residual 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
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.