Definition:Real Interval/Unbounded

Definition

Unbounded Closed Interval

There are two unbounded closed intervals involving a real number $a \in \R$, defined as:

 $\ds \hointr a \to$ $:=$ $\ds \set {x \in \R: a \le x}$ $\ds \hointl \gets a$ $:=$ $\ds \set {x \in \R: x \le a}$

Unbounded Open Interval

There are two unbounded open intervals involving a real number $a \in \R$, defined as:

 $\ds \openint a \to$ $:=$ $\ds \set {x \in \R: a < x}$ $\ds \openint \gets a$ $:=$ $\ds \set {x \in \R: x < a}$

Unbounded Interval without Endpoints

The unbounded interval without endpoints is equal to the set of real numbers:

$\openint \gets \to = \R$

Also known as

Some sources refer to these as infinite (real) intervals.