---
tag: 0TK8Y3J5RW
kind: definition
name: Interval
msc: [26A03]
defines: 0Q1D85B8G6
uses: [0337CYB0VD]
sources:
  - url: https://en.wikipedia.org/wiki/Interval_(mathematics)
    locator: Definition of an interval
  - url: https://en.wikipedia.org/wiki/Interval_(mathematics)
    locator: Notations for intervals
---

## Definition

A subset $I$ of $\mathbb{R}$ is an interval when, for all $x$ and $y$ in $I$ and each real $z$ with $x \le z \le y$, also $z \in I$.
For real numbers $a \le b$, the notation is:
$$[a, b] = \{x \in \mathbb{R} : a \le x \le b\}, \qquad (a, b) = \{x \in \mathbb{R} : a < x < b\},$$
$$[a, b) = \{x \in \mathbb{R} : a \le x < b\}, \qquad (a, b] = \{x \in \mathbb{R} : a < x \le b\}.$$
The set $[a, b]$ is a closed interval and $(a, b)$ is an open interval.
For each real $a$, the notation is $[a, \infty) = \{x : x \ge a\}$, $(a, \infty) = \{x : x > a\}$, $(-\infty, a] = \{x : x \le a\}$ and $(-\infty, a) = \{x : x < a\}$.

## Remarks

By the [[completeness axiom|0AXWET80YW]], each interval is one of the sets above, or $\mathbb{R}$, or $\emptyset$.
In some countries, the notation $]a, b[$ replaces $(a, b)$.
The symbols $\infty$ and $-\infty$ are not real numbers. They show only that the interval has no bound on that side.

## Explanations

### L1 human

An interval is a piece of the number line without holes.
A square bracket means that the end point belongs to the interval, and a round bracket means that it does not.
For example, $[0, 1)$ contains $0$ and $0.999$, but not $1$.