Closed interval

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Definition

We define a closed interval, denoted $[a,b]$, in $\mathbb{R}$ as follows:

• $[a,b]:\eq\left\{x\in\mathbb{R}\ \vert\ a\le x\le b\right\}$

We adopt the following conventions:

• if $a\eq b$ then $[a,b]$ is the singleton $\{a\}\subseteq\mathbb{R}$.
• if $b< a$ then $[a,b]:\eq\emptyset$

A closed interval in $\mathbb{R}$ is actually an instance of a closed ball in $\mathbb{R}$ based at $\frac{a+b}{2}$ and of radius $\frac{b-a}{2}$ - see claim 2 below.

A closed interval is called a "closed interval" because it is actually closed. See Claim 1 below

Generalisations

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There is a generalisation to a line between two points, including the points

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Proof of claims

Claim 1: The closed interval is closed

Recall a set is closed if its complement is open. The complement is $(-\infty,a)\cup(b,+\infty)$