Intersections of Open Sets and Unions of Closed Sets

The sets ${\mathcal {G}}^{\delta }$ and ${\mathcal {F}}^{\sigma }$ are a subset of the Borel set. The set ${\mathcal {G}}^{\delta }$ is the collection of all sets which are a countable intersections of open sets and the set ${\mathcal {F}}^{\sigma }$ is the collection of all sets which are a countable union of closed sets.

Definitions

Let X be a topological space whose collection of open sets is denoted ${\mathcal {G}}$ and whose collection of closed sets is denoted ${\mathcal {F}}$, then,

${\mathcal {G}}^{\delta }=\left\{\bigcap _{n=1}^{\infty }O_{n}:O_{n}\in {\mathcal {G}}~~\forall n\in \mathbb {N} \right\}$

${\mathcal {F}}^{\sigma }=\left\{\bigcup _{n=1}^{\infty }C_{n}:C_{n}\in {\mathcal {F}}~~\forall n\in \mathbb {N} \right\}$

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

${\mathcal {G}}^{w}=\left\{\bigcap _{n=1}^{\infty }P_{n}:P_{n}\in {\mathcal {G}}^{u}~~\forall n\in \mathbb {N} \right\}$

${\mathcal {F}}^{w}=\left\{\bigcap _{n=1}^{\infty }P_{n}:P_{n}\in {\mathcal {F}}^{u}~~\forall n\in \mathbb {N} \right\}$

if $w_{m}=\delta$ and we define

${\mathcal {G}}^{w}=\left\{\bigcup _{n=1}^{\infty }P_{n}:P_{n}\in {\mathcal {G}}^{u}~~\forall n\in \mathbb {N} \right\}$

${\mathcal {F}}^{w}=\left\{\bigcup _{n=1}^{\infty }P_{n}:P_{n}\in {\mathcal {F}}^{u}~~\forall n\in \mathbb {N} \right\}$

if $w_{m}=\sigma$.

Examples of ${\mathcal {G}}^{\delta }$ Sets

• For any topological space, X, any open set of X is in ${\mathcal {G}}^{\delta }$.

• Consider $\mathbb {R}$ under the standard topology. Then the set of irrational numbers is in ${\mathcal {G}}^{\delta }$.

• Let $f:\mathbb {R} \to \mathbb {R}$ be a function. Let $D\subset \mathbb {R}$ be the set of points for which $f$ is discontinuous. Then the complement of $D$ is in ${\mathcal {G}}^{\delta }$.

Properties of ${\mathcal {G}}^{\delta }$ Sets

• A set S is a ${\mathcal {G}}^{\delta }$ set if and only if its complement is a ${\mathcal {F}}^{\sigma }$ set

• ${\mathcal {G}}^{\delta }$ 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 $E$ is Lebesgue measurable iff it differs from a ${\mathcal {G}}^{\delta }$ by a set of measure zero. The backward direction is trivial, since ${\mathcal {G}}^{\delta }$ sets and sets of measure zero are both measurable. For the forward direction recall the we may find an open set ${\mathcal {O}}_{n}$ such that $\lambda ({\mathcal {O}}_{n}\setminus E)<{\frac {1}{n}}$. Then if we define $A=\cap {\mathcal {O}}_{n}$, clearly $A$ is a ${\mathcal {G}}^{\delta }$ set and by monotonicity $\lambda (A\setminus E)<{\frac {1}{n}}$ for all n, so $A\setminus E$ has measure 0.