Gδ set

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 in Germany with "G" for ' ("German": area) meaning open set in this case and δ for ' ("German": intersection). The term inner limiting set is also used. Gδ sets, and their dual Fσ sets, are the second level of the Borel hierarchy.

Definition

In a topological space a Gδ set is a countable intersection of open sets. The Gδ sets are exactly the level mathbf{Pi}^0_2 sets of the Borel hierarchy.

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.

Properties

A key property of Gδ sets is that they are the possible sets at which a function between metric spaces is continuous. Formally:

The set of points where a function "f" is continuous is a Gδ set.

This is because continuity at a point "p" can be defined by a Pi^0_2 formula, namely .The formula states that for every natural number epsilon > 0, there exists a natural number N > 0 such that whenever 0 < |x-p| < 1/N = delta, we have |f(x) - f(p)| < 1/epsilon. If you fix a value of epsilon, the set of "x" for which there is a corresponding "N" is an open set, and the universal quantifier on the epsilon corresponds to the intersection of these sets.

As a consequence, while it is possible for the irrationals to be the set of continuity points of a function (see the popcorn function), it is impossible to construct a function which is continuous only on the rational numbers.

Basic properties

* The complement of a Gδ 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 countable union of Gδ sets is called a Gδσ set.

* In metrizable spaces, every closed set is a Gδ set and, dually, every open set is an Fσ 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".

* A set that contains the intersection of a countable collection of dense open sets is called comeagre or residual. These sets are used to define generic properties of topological spaces of functions.

Gδ space

A G&delta; space is a topological space in which every closed set is a Gδ set.Fact|date=August 2008 A normal space which is also a G&delta; space is perfectly normal. Every metrizable space is perfectly normal, and every perfectly normal space is completely normal: neither implication is reversible.

See also

* Fσ set, the dual concept; note that "G" is German (') and "F" is French (').

References

* John L. Kelley, "General topology", van Nostrand, 1955. P.134.
* | year=1995 P. 162.


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