The algebra of finite disjoint unions of half-open intervals in $\mathbb{R}$ with extended endpoints

Call an h-interval any interval of one of the four forms

$(-\mathrm{\infty },b],(a,b],(a,\mathrm{\infty }),\mathbb{R},$

where $a<b$ are real in the bounded case. Let $\mathcal{H}$ be the family of subsets $E\subseteq \mathbb{R}$ for which there are a natural number $m$ and pairwise disjoint h-intervals $I_1,\dots ,I_m$ such that

$E={\bigcup }_{i=1}^m{I}_i\mathrm{.}$

The empty set is included as the empty union. This family is called the half-open interval algebra on $\mathbb{R}$.

Every h-interval is an interval in the sense of Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length, and the complement in $\mathbb{R}$ of one h-interval is empty, another h-interval, or a disjoint union of two h-intervals. Intersections of h-intervals are h-intervals or empty, so finite disjoint unions of h-intervals are closed under complements, finite unions, and finite intersections. Therefore $\mathcal{H}$ is an algebra of subsets of $\mathbb{R}$ in the sense of Algebras of subsets.