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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).

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

## Bounded intervals

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

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

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

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

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

{\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

• a left-bounded interval (right-unbounded interval), which is either
• a left-open interval (left-open ray or left-open half-line): $(a,+\infty )=\{x\,|\,x>a\}$ ,
• a left-closed interval (left-closed ray or left-closed half-line): $[a,+\infty )=\{x\,|\,x\geq a\}$ (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): $(-\infty ,b)=\{x\,|\,x ,
• a right-closed interval (right-closed ray or right-closed half-line): $(-\infty ,b]=\{x\,|\,x\leq b\}$ (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)

$(-\infty ,+\infty )={\mathopen {]}}\!-\infty ,+\infty {\mathclose {[}}=\mathbb {R} .$ The real line (continuum) has the same cardinality as the unit interval, since

$x\in (0,1)\mapsto \log \left({\frac {x}{1-x}}\right)\in \mathbb {R} ,$ $y\in \mathbb {R} \mapsto {\frac {1}{1+e^{-y}}}\in (0,1),$ which are bijective, i.e. one-to-one and onto, strictly increasing maps.