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Concept
Gδ set
Summary
DISPLAYTITLE:Gδ set
In the mathematical field of topology, a Gδ set is a subset of a topological space that is a countable intersection of open sets. The notation originated from the German nouns Gebiet and Durchschnitt .
Historically Gδ sets were also called inner limiting sets, but that terminology is not in use anymore.
Gδ sets, and their dual, Fsigma sets, are the second level of the Borel hierarchy.
In a topological space a Gδ set is a countable intersection of open sets. The Gδ sets are exactly the level Π sets of the Borel hierarchy.
Any open set is trivially a Gδ set.
The irrational numbers are a Gδ set in the real numbers . They can be written as the countable intersection of the open sets (the superscript denoting the complement) where is rational.
The set of rational numbers is a Gδ set in . If were the intersection of open sets each would be dense in because is dense in . However, the construction above gave the irrational numbers as a countable intersection of open dense subsets. Taking the intersection of both of these sets gives the empty set as a countable intersection of open dense sets in , a violation of the .
The continuity set of any real valued function is a Gδ subset of its domain (see the "Properties" section for a more general statement).
The zero-set of a derivative of an everywhere differentiable real-valued function on is a Gδ set; it can be a dense set with empty interior, as shown by Pompeiu's construction.
The set of functions in not differentiable at any point within contains a dense Gδ subset of the metric space . (See .)
The notion of Gδ sets in metric (and topological) spaces is related to the notion of completeness of the metric space as well as to the . See the result about completely metrizable spaces in the list of properties below. sets and their complements are also of importance in real analysis, especially measure theory.
The complement of a Gδ set is an Fσ set, and vice versa.
The intersection of countably many Gδ sets is a Gδ set.
The union of many Gδ sets is a Gδ set.
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