So I know that the extended real line is given by $\mathbb{R} \bigcup$ {$-\infty, \infty$}. So these are the facts that I know:
1) Firstly, every interval in $\mathbb{R}$ is a Borel Set (I seem to have read on another thread on stack exchange about how the axiom of choice gives us an interval which is NOT a borel set but I think we ignore that in this course and just assume every interval is indeed a Borel set).
2)Every Open set in $\mathbb{R}$ is a countable union of open intervals
3)The Borel $\sigma$-algebra of $\overline{\mathbb{R}}$ is generated by intervals of the form $(a, \infty]$ or $[a, \infty]$.
So my question is: Is every interval in $\overline{\mathbb{R}}$ also a borel set and is $(a,\infty]$ considered an open interval in $\overline{\mathbb{R}}$?
Best Answer
In $\overline{\mathbb R}$ intervals of the form $[-\infty,b)$ and $(a,\infty]$ are open. Actually, these intervals form by definition of the topology on $\overline{\mathbb R}$ bases of the neighbourhood system of the infinite points. Hence the complements $[a,\infty]$ and $[-\infty,b]$ are closed. Intervals in $\overline{\mathbb R}$ are of the form $(a,b)$ or $[a,b)$ or $(a,b]$ or $[a,b]$. The first is open, the last is closed, the other are intersections of an open and a closed interval, hence certainly all these are Borel.