Definition:Real Interval

The set of all real numbers between any two given real numbers $$a$$ and $$b$$ is called a (real) interval.

The numbers $$a, b \in \mathbb{R}$$ are known as the endpoints of the interval.

The difference $$\left|{a - b}\right|$$ between the endpoints is called the length of the interval.

An arbitrary interval is frequently denoted $$\mathbb{I}$$, although some sources use just $$I$$.

It is usual to define intervals in terms of inequalities.

These are in the form of a pair of brackets, either round or square, enclosing the two end points of the interval separated by two dots.

Whether the bracket at either end is round or square depends on whether the end point is inside or outside the interval, as specified in the following.

Suppose $$a, b \in \mathbb{R}$$.

Open Interval
The open interval from $$a$$ to $$b$$ is defined as:

$$\left ({a \, . \, . \, b} \right) = \left\{{x \in \reals: a < x < b}\right\}$$

Half Open Interval
There are two half open intervals from $$a$$ to $$b$$, defined as:


 * $$\left [{a \, . \, . \, b} \right) = \left\{{x \in \reals: a \le x < b}\right\}$$


 * $$\left ({a \, . \, . \, b} \right] = \left\{{x \in \reals: a < x \le b}\right\}$$

Closed Interval
The closed interval from $$a$$ to $$b$$ is defined as:

$$\left [{a \,. \, . \, b} \right] = \left\{{x \in \reals: a \le x \le b}\right\}$$

Unbounded Half Open Interval
There are two unbounded half open intervals involving a real number $$a$$, defined as:


 * $$\left [{a \, . \, . \, \infty} \right) = \left\{{x \in \reals: a \le x}\right\}$$


 * $$\left ({-\infty \, . \, . \, a} \right] = \left\{{x \in \reals: x \le a}\right\}$$

Unbounded Open Interval
There are two unbounded open intervals involving a real number $$a$$, defined as:


 * $$\left ({a \, . \, . \, \infty} \right) = \left\{{x \in \reals: a < x}\right\}$$


 * $$\left ({-\infty \, . \, . \, a} \right) = \left\{{x \in \reals: x < a}\right\}$$

Using the same symbology, the set $$\mathbb{R}$$ can be represented as an unbounded open interval with no end points:

$$\left ({-\infty \, . \, . \, \infty} \right) = \left\{{x \in \reals}\right\}$$

Notation
The notation as used here is a fairly recent innovation, and was introduced by C. A. R. Hoare and Lyle Ramshaw.

These are the notations usually seen for intervals:


 * $$\left ( {a, b} \right )$$ for $$\left ({a \, . \, . \, b} \right)$$;
 * $$\left [ {a, b} \right )$$ for $$\left [{a \, . \, . \, b} \right)$$;
 * $$\left ( {a, b} \right ]$$ for $$\left ({a \, . \, . \, b} \right]$$;
 * $$\left [ {a, b} \right ]$$ for $$\left [{a \, . \, . \, b} \right]$$.

... but they can be confused with other usages for these (in particular, we have the danger of taking $$\left({a, b}\right)$$ to mean an ordered pair and goodness knows what else).

Some authors try to get around this ambiguity problem by using the following notations for open and half-open intervals:


 * $$\left ] {a, b} \right [$$ for $$\left ({a \, . \, . \, b} \right)$$;
 * $$\left [ {a, b} \right [$$ for $$\left [{a \, . \, . \, b} \right)$$;
 * $$\left ] {a, b} \right ]$$ for $$\left ({a \, . \, . \, b} \right]$$.

... but these are both ugly and confusing, and not many people like those either.

The "double dots" notation used to denote an interval has a worthy precedent in the sphere of computer languages. For example, Pascal uses the same notation for a closed interval.

Higher Dimensional Intervals
An interval in $$\mathbb{R}^n$$ is the cartesian product:

$$\mathbb{I}_1 \times \mathbb{I}_2 \times \cdots \times \mathbb{I}_n$$

where $$\mathbb{I}_1, \ldots, \mathbb{I}_n$$ are intervals in $$\mathbb{R}$$.

Comment
Compare the definition of a Closed Interval on a general totally ordered set.

Note that only in the case of the closed interval are both endpoints actually included in the interval.