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