I am trying to understand the Borel Sigma Algebra on the extended real line. In order to do this, I must first understand the topology of the extended real line. In particular, I was wondering if $\{(a,b): a,b \in \mathbb{R} \}$ can generate (as in, use $\cup$ and complement of these to get all open sets) the extended real line topology. Since this set generates $\mathbb{R}$, we can find the complement of $\mathbb{R}$ and get $\{-\infty, \infty \}$, but I am not sure how to get the neighbourhoods of $\infty$ for example, $(a, \infty ]$.
[Math] Borel sigma algebra on Extended Real Line
borel-setsgeneral-topology
Best Answer
As you have discovered, the finite open intervals are not enough to generate the usual topology of the extended real line.
In order to generate the topology you need to explicitly declare $[-\infty,a)$ and $(a,\infty]$ (for $a\in\mathbb R$ or even just for $a\in\mathbb Q$ or $a\in \mathbb Z$) to be open.
(Remember that when you're generating a topology, taking complements is not a valid operation because complement is not supposed to preserve openness).
If all you're interested in is generate the Borel algebra, less can do it -- but you still need some way to distinguish between $-\infty$ and $\infty$. However including just, for example $\{-\infty\}$ or $[-\infty,0)$ among your generators would do the trick.