# Intersections of Open Sets and Unions of Closed Sets

The sets ${\displaystyle {\mathcal {G}}^{\delta }}$ and ${\displaystyle {\mathcal {F}}^{\sigma }}$ are a subset of the Borel set. The set ${\displaystyle {\mathcal {G}}^{\delta }}$ is the collection of all sets which are a countable intersections of open sets and the set ${\displaystyle {\mathcal {F}}^{\sigma }}$ is the collection of all sets which are a countable union of closed sets.[1]

## Definitions

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

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

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

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

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

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

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

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

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

if ${\displaystyle w_{m}=\sigma }$.

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

• For any topological space, X, any open set of X is in ${\displaystyle {\mathcal {G}}^{\delta }}$.
• Consider ${\displaystyle \mathbb {R} }$ under the standard topology. Then the set of irrational numbers is in ${\displaystyle {\mathcal {G}}^{\delta }}$.
• Let ${\displaystyle f:\mathbb {R} \to \mathbb {R} }$ be a function. Let ${\displaystyle D\subset \mathbb {R} }$ be the set of points for which ${\displaystyle f}$ is discontinuous. Then the complement of ${\displaystyle D}$ is in ${\displaystyle {\mathcal {G}}^{\delta }}$.

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

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

## References

1. Gerald B. Folland, Real Analysis: Modern Techniques and Their Applications, Second Edition, §1.2