Algebra Generated by Open and Closed Intervals

Question

If $E$ is the collection of all open intervals $(a,b)$ in $X=[0,1]$, how do I know that the $\sigma(E)$ contains all closed intervals $[a,b] \subset X$, in particular closed intervals involving the endpoints? (Regular closed intervals I can represent as countable intersections of open intervals, hence they are in $\sigma(E)$.)

Answer 1

Let $X = [0, 1]$.

If we define $E = \{(a, b) \cap X : a, b \in \mathbb R$ and $a < b\}$, then

• $(-1, a) \cap X = [0, a) \in E \subseteq \sigma(E)$ and
• $(b, 2) \cap X = (b, 1] \in E \subseteq \sigma(E)$.

So if we want $[a, b]$ in general, we observe that $(b, 1], [0, a) \in E \subseteq \sigma(E)$ and so are there complements: $[0, b], [a, 1]$.

Taking their intersection, we get $[a, b] \in \sigma(E)$ since $\sigma(E)$ is closed under intersections.