# Closed interval

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## Definition

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

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

• if [ilmath]a\eq b[/ilmath] then [ilmath][a,b][/ilmath] is the singleton [ilmath]\{a\}\subseteq\mathbb{R} [/ilmath].[Note 1]
• if [ilmath]b< a[/ilmath] then [ilmath][a,b]:\eq\emptyset[/ilmath]

A closed interval in [ilmath]\mathbb{R} [/ilmath] is actually an instance of a closed ball in [ilmath]\mathbb{R} [/ilmath] based at [ilmath]\frac{a+b}{2} [/ilmath] and of radius [ilmath]\frac{b-a}{2} [/ilmath] - 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