Closed interval
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Contents
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]
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
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 [ilmath](-\infty,a)\cup(b,+\infty)[/ilmath]
Notes
- ↑ 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.
