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# Intervals

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A (real) interval is a set of real numbers with the property that any number that lies between the lower bound (when left-bounded) and the upper bound (when right-bounded) of the set is also included in the set, thus it is connected (which implies convex, since it is one-dimensional).

## Lower and upper bounds

The lower bound is called either

• the minimum (if included),
• the infimum (if excluded),

while the upper bound is called either

• the maximum (if included);
• the supremum (if excluded).

An interval is left-closed, i.e. contains all its limit points to the left, if it is either

• a left-unbounded interval or the unbounded interval (since it contains all its limit points to the left);
• a left-closed [right-unbounded or bounded] interval.

An interval is right-closed, i.e. contains all its limit points to the right, if it is either

• a right-unbounded interval or the unbounded interval (since it contains all its limit points to the right);
• a right-closed [left-unbounded or bounded] interval.

A closed interval is left-closed and right-closed, i.e. contains all its limit points.

An open interval is left-open and right-open.

## Degenerate intervals

A degenerate interval has its upper bound equal to its lower bound and is either

• a singleton ${\displaystyle [a,a]=\{a\}}$;
• the empty set ${\displaystyle (a,a)=\emptyset }$.[1]

## Bounded intervals

Notation (standard notation in the left column, nonstandard notation in the middle column) for bounded intervals:

{\displaystyle {\begin{aligned}(a,b)={\mathopen {]}}a,b{\mathclose {[}}&=\{x\in \mathbb {R} \,|\,a

A bounded interval is

• an open interval, denoted ${\displaystyle (a,b)}$, which is the set of all real numbers between ${\displaystyle a}$ and ${\displaystyle b}$, exclusively;
• an open-closed half-open interval (resp., half-closed interval), denoted ${\displaystyle (a,b]}$, which excludes ${\displaystyle a}$;[2]
• a closed-open half-open interval (resp., half-closed interval), denoted ${\displaystyle [a,b)}$, which excludes ${\displaystyle b}$;[2] or
• a closed interval, denoted ${\displaystyle [a,b]}$, which is the set of all real numbers from ${\displaystyle a}$ to ${\displaystyle b}$, inclusively.

All [nondegenerate] bounded intervals have the same cardinality as the unit interval, since we have the linear maps

${\displaystyle x\in (0,1)\mapsto a+x\,(b-a)\in (a,b),}$
${\displaystyle y\in (a,b)\mapsto {\frac {y-a}{b-a}}\in (0,1),}$

which are bijective, i.e. one-to-one and onto.

## Half-bounded intervals

Notation (standard notation in the left column, nonstandard notation in the middle column) for half-bounded intervals:

{\displaystyle {\begin{aligned}(a,+\infty )={\mathopen {(}}a,+\infty {\mathclose {[}}&=\{x\in \mathbb {R} \,|\,a

A half-bounded interval (half-unbounded interval), also called ray or half-line, is either[3]

• a left-bounded interval (right-unbounded interval), which is either
• a left-open interval (left-open ray or left-open half-line): ${\displaystyle (a,+\infty )=\{x\,|\,x>a\}}$,
• a left-closed interval (left-closed ray or left-closed half-line): ${\displaystyle [a,+\infty )=\{x\,|\,x\geq a\}}$ (a closed interval, since it contains all its limit points);
• a left-unbounded interval (right-bounded interval), which is either
• a right-open interval (right-open ray or right-open half-line): ${\displaystyle (-\infty ,b)=\{x\,|\,x,
• a right-closed interval (right-closed ray or right-closed half-line): ${\displaystyle (-\infty ,b]=\{x\,|\,x\leq b\}}$ (a closed interval, since it contains all its limit points).

## Unbounded interval

The unbounded interval is the set of real numbers itself (a closed interval, since it contains all its limit points)[3]

${\displaystyle (-\infty ,+\infty )={\mathopen {]}}\!-\infty ,+\infty {\mathclose {[}}=\mathbb {R} .}$

The real line (continuum) has the same cardinality as the unit interval, since

${\displaystyle x\in (0,1)\mapsto \log \left({\frac {x}{1-x}}\right)\in \mathbb {R} ,}$
${\displaystyle y\in \mathbb {R} \mapsto {\frac {1}{1+e^{-y}}}\in (0,1),}$

which are bijective, i.e. one-to-one and onto, strictly increasing maps.

## Notes

1. Also the empty set, since all real numbers are finite: ${\displaystyle (-\infty ,-\infty )=\emptyset }$ and ${\displaystyle (+\infty ,+\infty )=\emptyset }$.
2. Sometimes the terms semi-open or semi-closed might be seen.
3. Here, the infinity symbol ${\displaystyle \infty }$ does not refer to the point at infinity (ideal infinity, belonging to the real projective line ${\displaystyle \mathbb {R} P^{1}}$), which does not belong to ${\displaystyle \mathbb {R} }$, it means unbounded, e.g. ${\displaystyle -\infty }$ means negatively (complex argument is ${\displaystyle \pi }$) unbounded and ${\displaystyle +\infty }$ (the ${\displaystyle +}$ sign is usually omitted) means positively (complex argument is ${\displaystyle 0}$) unbounded.