Closed interval

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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]

We adopt the following conventions:

  • 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


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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 [ilmath](-\infty,a)\cup(b,+\infty)[/ilmath]


  1. Effectively this is [ilmath][a,a][/ilmath] or [ilmath][b,b][/ilmath]. It is easy to see that [ilmath]\{x\in\mathbb{R}\ \vert\ a\le x\le a\} [/ilmath] is just [ilmath]x\eq a[/ilmath] itself.