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# Intervals

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).

## Contents

## 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 ;
- the empty set .
^{[1]}

## Bounded intervals

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

A *bounded interval* is

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

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

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:

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*): , - a
*left-closed interval*(*left-closed ray*or*left-closed half-line*): (a closed interval, since it contains all its limit points);

- a
- a
*left-unbounded interval*(*right-bounded interval*), which is either- a
*right-open interval*(*right-open ray*or*right-open half-line*): , - a
*right-closed interval*(*right-closed ray*or*right-closed half-line*): (a closed interval, since it contains all its limit points).

- a

## Unbounded interval

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

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

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

## See also

## Notes

- ↑ Also the empty set, since all real numbers are finite: and .
- ↑
^{2.0}^{2.1}Sometimes the terms**semi-open**or**semi-closed**might be seen. - ↑
^{3.0}^{3.1}Here, the infinity symbol does not refer to the point at infinity (ideal infinity, belonging to the real projective line ), which does not belong to , it means unbounded, e.g. means negatively (complex argument is ) unbounded and (the sign is usually omitted) means positively (complex argument is ) unbounded.

## External links

- Weisstein, Eric W., Interval, from MathWorld—A Wolfram Web Resource.