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Difference between revisions of "Closed interval"

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A '''closed interval''' is an [[interval]] which has both a [[maximum]] and a [[minimum]] element. Closed intervals are denoted by <math>[a,b]</math> where <math>a</math> is the [[minimum]] and <math>b</math> is the [[maximum]]. In the interval <math>[a,b]</math>, <math>a</math> is also the [[Greatest lower bound | infimum]] (greatest lower bound) and <math>b</math> is also the [[Least upper bound | supremum]] (least upper bound). Alternatively, a closed interval is the [[set]] of all <math>x</math> such that <math>x</math> satisfies both of the inequalities <math>a \leq x</math> and <math>x \leq b</math>.
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A '''closed interval''' is an [[interval]] which has both a [[maximum]] and a [[minimum]] element. Closed intervals are denoted by <math>[a,b]</math> where <math>a</math> is the [[minimum]] and <math>b</math> is the [[maximum]]. In the interval <math>[a,b]</math>, <math>a</math> is also the [[Greatest lower bound | infimum]] (greatest lower bound) and <math>b</math> is also the [[Least upper bound | supremum]] (least upper bound). Alternatively, a closed interval <math>[a,b]</math> is the [[set]] of all <math>x</math> such that <math>x</math> satisfies both of the inequalities <math>a \leq x</math> and <math>x \leq b</math>.
  
 
Every closed interval is a [[closed set]].
 
Every closed interval is a [[closed set]].

Latest revision as of 13:59, 5 March 2022

A closed interval is an interval which has both a maximum and a minimum element. Closed intervals are denoted by $[a,b]$ where $a$ is the minimum and $b$ is the maximum. In the interval $[a,b]$, $a$ is also the infimum (greatest lower bound) and $b$ is also the supremum (least upper bound). Alternatively, a closed interval $[a,b]$ is the set of all $x$ such that $x$ satisfies both of the inequalities $a \leq x$ and $x \leq b$.

Every closed interval is a closed set.

Examples

$[-1,1]$, the set of real numbers nonstrictly between $-1$ and $1$, is a closed interval.

For any real number $x$, the set $\{x\} = [x,x]$ containing only $x$ is a closed interval.

See also

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