Difference between revisions of "Open interval"

Latest revision as of 16:20, 19 September 2022

An open interval is an interval which has neither a maximum nor a minimum element. Open intervals are denoted by $(a,b)$ where $a$ is the infimum (greatest lower bound) and $b$ is the supremum (least upper bound). Alternatively, an open interval $(a,b)$ is the set of all $x$ such that $x$ satisfies both of the inequalities $a < x$ and $x < b$ .

In an open interval, it is possible that either the infimum $a$ or the supremum $b$ , or both, do not exist. If the infimum is nonexistent, the value of $a$ is written as $-\infty$ ; if the supremum is nonexistent, the value of $b$ is written as $\infty$ . The corresponding inequality is always considered true in these cases, since $-\infty < x$ and $x < +\infty$ by definition.

Every open interval is an open set.

Examples

$\mathbb R$ , the set of all real numbers, is an open interval with neither an upper bound nor a lower bound.

$(0, + \infty)$ , the set of positive real numbers, is an open interval with a lower bound but no upper bound.

$(-\infty, 0)$ , the set of negative real numbers, is an open interval with an upper bound but no lower bound.

$(-1, 1)$ , the set of real numbers strictly between $-1$ and $1$ , is an open interval with both an upper bound and a lower bound.

The empty set, having no elements and therefore neither a maximum nor a minimum, is considered an open interval.

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-An '''open interval''' is an [[interval]] which has neither a [[maximum]] nor a [[minimum]] element. Open intervals are denoted by <math>(a,b)</math> where <math>a</math> is the [[Greatest lower bound | infimum]] (greatest lower bound) and <math>b</math> is the [[Least upper bound | supremum]] (least upper bound). Alternatively, an open interval is the [[set]] of all <math>x</math> such that <math>x</math> satisfies both of the [[Inequality|inequalities]] <math>a < x</math> and <math>x < b</math>.
+An '''open interval''' is an [[interval]] which has neither a [[maximum]] nor a [[minimum]] element. Open intervals are denoted by <math>(a,b)</math> where <math>a</math> is the [[Greatest lower bound | infimum]] (greatest lower bound) and <math>b</math> is the [[Least upper bound | supremum]] (least upper bound). Alternatively, an open interval <math>(a,b)</math> is the [[set]] of all <math>x</math> such that <math>x</math> satisfies both of the [[Inequality|inequalities]] <math>a < x</math> and <math>x < b</math>.
-In an open interval, it is possible that either <math>a</math> or <math>b</math>, or both, do not exist. If <math>a</math> is nonexistent, the value of <math>a</math> is written as <math>-\infty</math>; if b is nonexistent, the value of <math>b</math> is written as <math>\infty</math>. The corresponding inequality is always considered true in these cases, since <math>-\infty < x</math> and <math>x < +\infty</math> by definition.
+In an open interval, it is possible that either the infimum <math>a</math> or the supremum <math>b</math>, or both, do not exist. If the infimum is nonexistent, the value of <math>a</math> is written as <math>-\infty</math>; if the supremum is nonexistent, the value of <math>b</math> is written as <math>\infty</math>. The corresponding inequality is always considered true in these cases, since <math>-\infty < x</math> and <math>x < +\infty</math> by definition.
 Every open interval is an [[open set]].

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Difference between revisions of "Open interval"

Latest revision as of 16:20, 19 September 2022

Examples

See also