Difference between revisions of "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 <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. | ||
Revision as of 13:59, 5 March 2022
An open interval is an interval which has neither a maximum nor a minimum element. Open intervals are denoted by 
where 
is the infimum (greatest lower bound) and 
is the supremum (least upper bound). Alternatively, an open interval is the set of all 
such that satisfies both of the inequalities 
and 
.
In an open interval, it is possible that either or , or both, do not exist. If is nonexistent, the value of is written as 
; if b is nonexistent, the value of is written as 
. The corresponding inequality is always considered true in these cases, since 
and 
by definition.
Every open interval is an open set.
Examples
, the set of all real numbers, is an open interval with neither an upper bound nor a lower bound.
, the set of positive real numbers, is an open interval with a lower bound but no upper bound.
, the set of negative real numbers, is an open interval with an upper bound but no lower bound.
, the set of real numbers strictly between 
and 
, 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.
See also
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