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The sets '''<math>\mathcal{G}^\delta</math>''' and '''<math>\mathcal{F}^\sigma</math>''' are a subset of the Borel set. The set '''<math>\mathcal{G}^\delta</math>''' is the collection of all sets which are a countable intersections of open sets and the set '''<math>\mathcal{F}^\sigma</math>''' is the collection of all sets which are a countable union of closed sets.<ref name="Folland1">Gerald B. Folland, ''Real Analysis: Modern Techniques and Their Applications, Second Edition'', §1.2 </ref>

==Definitions==

Let ''X'' be a topological space whose collection of open sets is denoted ''<math>\mathcal{G}</math>'' and whose collection of closed sets is denoted ''<math>\mathcal{F}</math>'', then,

<math>\mathcal{G}^\delta = \left \{\bigcap_{n=1}^\infty O_n: O_n\in\mathcal{G}~~ \forall n\in\mathbb{N} \right \}</math>

<math>\mathcal{F}^\sigma = \left \{\bigcup_{n=1}^\infty C_n: C_n\in \mathcal{F}~~ \forall n\in\mathbb{N} \right \}</math>

The definitions can be extended as follows. Let ''w'' be a non-trivial word in the alphabet <math>\{\delta,\sigma\}</math> of length ''m''. Let ''u'' be the first ''m-1'' letters in the word and let ''<math>w_m</math>'' be the last letter. Then we define,

<math>\mathcal{G}^w = \left \{\bigcap_{n=1}^\infty P_n: P_n\in \mathcal{G}^u~~ \forall n\in\mathbb{N} \right \}</math>

<math>\mathcal{F}^w = \left \{\bigcap_{n=1}^\infty P_n: P_n\in \mathcal{F}^u~~ \forall n\in\mathbb{N} \right \}</math>

if <math>w_m = \delta</math> and we define

<math>\mathcal{G}^w = \left \{\bigcup_{n=1}^\infty P_n: P_n\in \mathcal{G}^u~~ \forall n\in\mathbb{N} \right \}</math>

<math>\mathcal{F}^w = \left \{\bigcup_{n=1}^\infty P_n: P_n\in \mathcal{F}^u~~ \forall n\in\mathbb{N} \right \}</math> 

if <math>w_m = \sigma</math>.

==Examples of ''<math>\mathcal{G}^\delta</math>'' Sets==

*For any topological space, ''X'', any open set of X is in <math>\mathcal{G}^\delta</math>.

*Consider <math>\mathbb{R}</math> under the standard topology. Then the set of irrational numbers is in <math>\mathcal{G}^\delta</math>.

*Let ''<math>f: \mathbb{R}\to\mathbb{R}</math>'' be a function. Let <math>D\subset\mathbb{R}</math> be the set of points for which <math>f</math> is discontinuous. Then the complement of <math>D</math> is in <math>\mathcal{G}^\delta</math>.

==Properties of ''<math>\mathcal{G}^\delta</math>'' Sets==

*A set ''S'' is a <math>\mathcal{G}^\delta</math> set if and only if its complement is a <math>\mathcal{F}^\sigma</math> set

*''<math>\mathcal{G}^\delta</math>'' is closed under finite union and countable intersection
* Lesbague measurable sets can be thought of as the completion of the Borel sets in the following way. A  set <math> E </math> is Lebesgue measurable iff it differs from a <math> \mathcal{G}^\delta </math> by a set of measure zero. The backward direction is trivial, since <math> \mathcal{G}^\delta </math> sets and sets of measure zero are both measurable. For the forward direction recall the we may find an open set <math> \mathcal{O}_n </math> such that <math> \lambda (\mathcal{O}_n\setminus E)<\frac{1}{n} </math>. Then if we define <math> A=\cap \mathcal{O}_n </math>, clearly <math> A </math> is a <math> \mathcal{G}^\delta </math> set and by monotonicity <math> \lambda(A\setminus E)<\frac{1}{n} </math> for all n, so <math> A \setminus E </math> has measure 0.

==References==