Prove that every open subset of $\mathbb R$ is an $F_\sigma$-set and a $G_\delta$-set.
In order to prove this, the exercise tells me I need to make use of the following previously proven facts:
- Every open interval in $\mathbb R$ is both $F_\sigma$-set and $G_\delta$-set.
- $\mathcal B = \{ (a,b) \in \mathbb R: a<b~ \text{and}~ a,b, \in \mathbb Q \}$ is a basis for the Euclidean topology on $\mathbb R$.
I have done the following so far:
We have proven that every open interval $(a,b) \subset \mathbb R$ is both a $F_\sigma$-set and $G_\delta$-set. We now attempt to prove that every open subset $S \subset \mathbb R$ is also both a $F_\sigma$-set and $G_\delta$-set.
Since $S$ is open in $\mathbb R$, we know that it is a union of open intervals, that is
\begin{align*}
S &= \bigcup_{j \in J} (a_j, b_j) \\
&= \bigcup_{j \in J} \left[ \bigcup_{n=1}^\infty \left[ a_j + \frac{1}{n} , b_j – \frac{1}{n}\right]\right],
\end{align*} which is a countable union of closed sets. That is, $S$ is an $F_\sigma$-set.
I now need help proving that $S$ is also a $G_\delta$-set. I am asssuming this will make use of (2) that they told us to use. I cannot seem to see how to do this though?
Best Answer
It is trival. Since $$ S=\bigcap_{n=1}^{\infty}S \quad\text{where }\quad S=\bigcup_{j \in J} (a_j, b_j) $$ S is $G_{\delta}$ for $S$ is open.