G-delta set
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In the mathematical field of topology a G-delta set or Gδ set is a set in a topological space which is in a certain sense simple. The notation originated in Germany with G for Gebiet (German:area) meaning open set in this case and δ for Durchschnitt (german:intersection).
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[edit] Definition
In a topological space a Gδ set is a countable intersection of open sets.
[edit] Examples
- Any open set is trivially a Gδ set
- The irrational numbers are a Gδ set in R, the real numbers, as they can be written as the intersection over all rational numbers q of the complement of {q} in R.
- The rational numbers Q are not a Gδ set. If we were able to write Q as the intersection of open sets An, each An would have to be dense in R since Q is dense in R. 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 R, a violation of the Baire category theorem.
- The set of points at which a function from R to itself is continuous can be shown to be a Gδ.
Thus while it may be possible for the irrationals to be the set of continuity points of a function (in fact, such a function does exist), it is impossible to construct a function which is continuous only on the rational numbers.
[edit] Properties
- In metrizable spaces, every closed set is a Gδ set.
- The complement of a Gδ set is an Fσ. In a metrizable space, every open set is an Fσ set.
- The intersection of countably many Gδ sets is a Gδ set, and the union of finitely many Gδ sets is a Gδ set.
- A subspace A of a topologically complete space X is itself topologically complete if and only if A is a Gδ set in X
[edit] See also
- F-sigma set, the dual concept
