# Countable union of closed sets/countable interesection of open sets

• Oct 8th 2010, 11:13 AM
Zalren
Countable union of closed sets/countable interesection of open sets
A set $\displaystyle A$ is called an $\displaystyle F_\sigma$ set if it can be written as the countable union of closed sets. A set $\displaystyle B$ is called a $\displaystyle G_\delta$ set if it can be written as the countable intersection of open sets.
(a) Show that a closed interval [a, b] is a $\displaystyle G_\delta$ set.
(b) Show that the half open interval (a, b] is both a $\displaystyle G_\delta$ and an $\displaystyle F_\sigma$ set.
(c) Show that $\displaystyle Q$ is an $\displaystyle F_\sigma$ set, and the set of irrationals $\displaystyle I$ forms a $\displaystyle G_\delta$ set.

I have no idea where to start on this problem. Any help would be appreciated. Thank you for your time.
• Oct 8th 2010, 11:21 AM
TheEmptySet
For (a) consider the family of open sets

$\displaystyle \displaystyle A_n=\left( a-\frac{1}{n},b+\frac{1}{n}\right)$

What happens when you take the intersection of

$\displaystyle \displaystyle \cap_{n=1}^{\infty}A_n$?

This should get you started on (a) and (b)

Also remember that in $\displaystyle \mathbb{R}$ points are closed and $\displaystyle \mathbb{Q}$ is a countable set.

Good luck.
• Oct 8th 2010, 01:11 PM
Plato
I have decided to show you the irrational part of part #3.
Let $\displaystyle \mathbb{I}=\mathbb{R}\setminus\mathbb{Q}$ the irrational numbers.

The let $\displaystyle \{x\}^c=\left( { - \infty ,x} \right) \cup \left( {x,\infty } \right)$.

Try to convince yourself that $\displaystyle \mathbb{I}= \bigcap\limits_{x \in \mathbb{Q}} {\left\{ x \right\}^c }$, a $\displaystyle G_{\delta}$ set.
• Oct 8th 2010, 01:59 PM
Zalren
Thanks all! You guys rock!