Intervals

Why

We name and denote subsets of the set of real numbers which correspond to segments of a line.

Definition

Take two real numbers, with the first less than the second.

An interval is one of four sets:

1. the set of real numbers larger than the first number and smaller than the second; we call the interval open.
2. the set of real numbers larger than or equal to the first number and smaller than or equal to the second number; we call the interval closed.
3. the set of real numbers larger than the first number and smaller than or equal to the second; we call the interval open on the left and closed on the right
4. the set of real numbers larger than or equal to the first number and smaller than the second; we call the interval closed on the left and open on the right.

We call the two numbers the endpoints of the interval. An open interval does not contain its endpoints. A closed interval contains its endpoints. A half-open/half-closed interval contains only one of its endpoints. We say that the endpoints delimit the interval.

Notation

Let $a, b$ be two real numbers which satisfy the relation $a < b$.

We denote the open interval from $a$ to $b$ by $\oi{a, b}$. This notation, although standard, is the same as that for ordered pairs; no confusion arises with adequate context.¹

We denote the closed interval from $a$ to $b$ by $\ci{a, b}$. We record the fact $\oi{a, b} \subset \ci{a, b}$ in our new notation.

We denote the half-open interval from $a$ to $b$, closed on the right, by $\oci{a, b}$ and the half-open interval from $a$ to $b$, closed on the left, by $\coi{a, b}$.²

The unit interval is the set $\ci{0_{\R }, 1_{\R }}$ and we sometimes denote it by $\I$.

1. In future editions, we may use $\oleft(a, b\oright)$ or even $\oleft[\,a, b\,\oright]$. ↩︎
2. Some authors use ]a, b], [a, b[ and ]a, b[. ↩︎