Intervals of the Real Line

In the setting of Routine Facts Used Without Citation: Arithmetic, Maps, Predicativity and Indices, let $a,b\in\mathbb{R}$.

(i) A subset $I\subseteq\mathbb{R}$ is an interval if $y\in I$ for all $x,y,z\in\mathbb{R}$ with $x,z\in I$ and $x\le y\le z$.

(ii) The closed interval $[a,b]$, the open interval $(a,b)$ and the half-open intervals $[a,b)$ and $(a,b]$ are the subsets

$$[a,b]=\{x\in\mathbb{R}:a\le x\le b\},\qquad (a,b)=\{x\in\mathbb{R}:a<x<b\},\qquad [a,b)=\{x\in\mathbb{R}:a\le x<b\},\qquad (a,b]=\{x\in\mathbb{R}:a<x\le b\}$$

of $\mathbb{R}$.

(iii) The rays are the subsets

$$(a,\infty)=\{x\in\mathbb{R}:a<x\},\qquad [a,\infty)=\{x\in\mathbb{R}:a\le x\},\qquad (-\infty,b)=\{x\in\mathbb{R}:x<b\},\qquad (-\infty,b]=\{x\in\mathbb{R}:x\le b\}$$

of $\mathbb{R}$. In this notation $\infty$ and $-\infty$ are symbols only; they denote no element of $\mathbb{R}$ or of any set.