Interval (in ℝ)

A subset $I\subseteq\mathbb{R}$ is an interval if

i.e. whenever $x,y\in I$ and $x<y$, every real number between them is also in $I$.

Intervals are exactly the connected subsets of $\mathbb{R}$ (with the usual topology), and they form the natural domains for one-variable analysis (limits, derivatives, integrals).

Examples:

• Open intervals $(a,b)$, closed intervals $[a,b]$, and half-open intervals $[a,b)$ are intervals.
• Rays $(a,\infty)$ and $(-\infty,b]$ are intervals.
• The set $(0,1)\cup(2,3)$ is not an interval.