Basic Facts about Intervals of the Real Line and Their Interior Points

Let $\mathbb{R}$ be the real numbers, with the order $\le$ of its ordered field structure and the associated strict order $<$, and write $s-t$ for $s+(-t)$. Let intervals and closed intervals $[a,b]$ be as in that definition, open intervals $(p,q)$ as in that definition, and interior points as in that definition.

Let $I\subseteq\mathbb{R}$ be an interval and let $a,b,p,q,u,v,x\in\mathbb{R}$. Then the following hold.

1. (The real line) ¶ $\mathbb{R}$ is an interval, it contains at least two points, and every $x\in\mathbb{R}$ is an interior point of $\mathbb{R}$.

2. (Closed subintervals) ¶ If $u,v\in I$ and $u\le v$, then $[u,v]\subseteq I$.

3. (Points strictly between) ¶ If $p,q\in I$ and $p<x<q$, then $x\in I$ and $x$ is an interior point of $I$.

4. (Open intervals) ¶ $(p,q)$ is an interval, and every $x\in(p,q)$ is an interior point of $(p,q)$.

5. (Closed intervals) ¶ If $a\le b$, then $[a,b]$ is an interval. If moreover $a<b$, then $[a,b]$ contains at least two points, and every $x\in(a,b)$ is an interior point of $[a,b]$.