Notes:Just what is in a generated sigma-algebra

Consider a topology on $X$, call it $\mathcal{J}$, what can we say about the $\sigma$-algebra generated by $\mathcal{J}$, or $\sigma(\mathcal{J})$ as it is called?

Notes about generation

We know from $\mathcal{J}\subseteq\sigma(\mathcal{J})$ that automatically (by the implies-subset relation) every open set is in $\sigma(\mathcal{J})$. We also know that $\sigma(\mathcal{J})$ is minimal, as small as it can be. This means:

• $\sigma(\mathcal{J})$ only contains things we can get to via the operations a $\sigma$-algebra allows starting from the open sets, $\mathcal{J}$

Example

Take the usual topology on $\mathbb{R}$ (which we will call $(\mathbb{R},\mathcal{J})$) generated by open sets of the form $(a,b)$ (as the interval $\{x\in\mathbb{R}\vert\ a< x< b\}$), and let us ask the question:

• Is $\{5\}\in\sigma(\mathbb{R})$? The answer is "yes", there are several ways.
  1. First we can do the famous counter example as to why toplogies require closure under finite intersection only:

     Let $$A=\bigcap_{n=1}^\infty\left(5-\tfrac{1}{n},5+\tfrac{1}{n}\right)$$ , it is easy to see that

     • We have used the $\sigma$-$\cap$-closed property, as we know that $\forall n\in\mathbb{N}[(5-\tfrac{1}{n},5+\tfrac{1}{n})\in\mathcal{J}\implies(5-\tfrac{1}{n},5+\tfrac{1}{n})\in\sigma(\mathcal{J})]$