Interval Notation
Interval notation is a way to describe a set of numbers on the number line using brackets and parentheses to indicate whether endpoints are included or excluded.
Definition
Interval notation is a shorthand way to write a range of numbers: square brackets $[ ]$ mean the endpoint is included (closed), and round brackets/parentheses $( )$ mean the endpoint is not included (open); infinity always uses a parenthesis, since you can never reach it, and the union symbol ($\cup$) connects disjoint intervals. Formally, an interval in $\mathbb{R}$ is a connected subset: a set $I$ such that for any $a, b \in I$ and any $c$ with $a < c < b$, $c \in I$. There are nine types of intervals: $(a,b)$, $[a,b]$, $(a,b]$, $[a,b)$, $(a,\infty)$, $[a,\infty)$, $(-\infty,b)$, $(-\infty,b]$, and $(-\infty,\infty) = \mathbb{R}$, and intervals are exactly the connected subsets of $\mathbb{R}$ under the standard topology.
Example
$[3, 7]$ means all numbers from $3$ to $7$, including $3$ and $7$, while $(3, 7)$ excludes both endpoints. $x \ge -4$ is written $[-4, \infty)$, $-2 < x \le 5$ is written $(-2, 5]$, and $x < 1$ or $x > 3$ is written $(-\infty, 1) \cup (3, \infty)$. The solution set of $x^2 < 9$ is $\{x : -3 < x < 3\} = (-3, 3)$, an open interval that is connected and open in the standard topology on $\mathbb{R}$.
Key Insight
Always use parentheses with infinity, since you can never reach it. Interval notation is more compact than set-builder notation and more informative than just writing the inequality, since it clearly shows boundedness and whether endpoints are included. The Heine-Borel theorem characterizes compact subsets of $\mathbb{R}$ as exactly the closed bounded intervals $[a, b]$, underpinning why closed intervals guarantee the existence of maxima and minima for continuous functions, a fact essential throughout analysis.