Definition:Real Interval

Definition

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

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

Endpoints

The numbers $a, b \in \R$ are known as the endpoints (or end points) of the interval.

$a$ is sometimes called the left hand endpoint and $b$ the right hand end point of the interval.

Length

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

Midpoint

The midpoint of an interval is the number $\dfrac {a + b} 2$.

Property Defining an Interval

An interval has the property $\forall x, y \in \mathbb I, x \le z \le y \implies z \in \mathbb I$.

That is, if two numbers belong to an interval, then so does every number in between them.

This is proved in Interval Defined by Betweenness.

Definitions of Interval Types

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 endpoints 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 $a, b \in \R$.

Open Interval

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

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

Half Open Interval

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

• $\left [{a . . b} \right) = \left\{{x \in \R: a \le x < b}\right\}$

• $\left ({a . . b} \right] = \left\{{x \in \R: a < x \le b}\right\}$

Closed Interval

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

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

Such an interval can also be referred to as compact.

Unit Interval

Some authors use $\mathbf I$ specifically to mean the (closed) unit interval, that is, the closed interval from $0$ to $1$:

$\left [{0 . . 1} \right] = \left\{{x \in \R: 0 \le x \le 1}\right\}$

When referring to the unit interval, it is usually understood that the closed unit interval $\left [{0 . . 1} \right]$ is meant.

Otherwise, the interval $\left ({0 . . 1} \right)$ is referred to as the open unit interval.

Unbounded Half Open Interval

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

• $\left [{a . . \infty} \right) = \left\{{x \in \R: a \le x}\right\}$

• $\left ({-\infty . . a} \right] = \left\{{x \in \R: x \le a}\right\}$

Unbounded Open Interval

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

• $\left ({a . . \infty} \right) = \left\{{x \in \R: a < x}\right\}$

• $\left ({-\infty . . a} \right) = \left\{{x \in \R: x < a}\right\}$

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

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

Empty Interval

When $a > b$, we have:

• $\left [{a . . b} \right] = \left\{{x \in \R: a \le x \le b}\right\} = \varnothing$

• $\left [{a . . b} \right) = \left\{{x \in \R: a \le x < b}\right\} = \varnothing$

• $\left ({a . . b} \right] = \left\{{x \in \R: a < x \le b}\right\} = \varnothing$

• $\left ({a . . b} \right) = \left\{{x \in \R: a < x < b}\right\} = \varnothing$

When $a = b$, we have:

• $\left [{a . . b} \right) = \left [{a . . a} \right) = \left\{{x \in \R: a \le x < a}\right\} = \varnothing$

• $\left ({a . . b} \right] = \left ({a . . a} \right] = \left\{{x \in \R: a < x \le a}\right\} = \varnothing$

• $\left ({a . . b} \right) = \left ({a . . a} \right) = \left\{{x \in \R: a < x < a}\right\} = \varnothing$

Singleton Interval

When $a = b$, we have:

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

Higher Dimensional Intervals

An interval in $\R^n$ 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 $\R$.

The interval $\R^n$ is called an n-dimensional interval.

Real Number Line as a Metric Space

From Real Number Line is Metric Space, one can define an open interval in terms of an $\epsilon$-neighborhood.

Thus any open interval $\left ({a . . b} \right)$ can be expressed as:

$\left ({\alpha - \epsilon . . \alpha + \epsilon} \right)$

where $\alpha = \dfrac {a + b} 2$ and $\epsilon = \dfrac {b - a} 2$.

Hence $\left ({\alpha - \epsilon . . \alpha + \epsilon} \right)$ is the $\epsilon$-neighborhood $N_\epsilon \left({\alpha}\right)$.

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.

This 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.

Alternative Notation for Unbounded Intervals

Some authors (sensibly, perhaps) prefer not to use the $\infty$ symbol and instead use:

	\(\displaystyle \)	\(\displaystyle \left [{a, \to} \right]\)	\(=\)	\(\displaystyle \left\{ {x \in \R: a \le x}\right\}\)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \left [{\gets, a} \right]\)	\(=\)	\(\displaystyle \left\{ {x \in \R: x \le a}\right\}\)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \left ]{a, \to} \right]\)	\(=\)	\(\displaystyle \left\{ {x \in \R: a < x}\right\}\)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \left [{\gets, a} \right[\)	\(=\)	\(\displaystyle \left\{ {x \in \R: x < a}\right\}\)	\(\displaystyle \)

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.

Sources

• James M. Hyslop: Infinite Series (1942): $\S 2$
• George McCarty: Topology: An Introduction with Application to Topological Groups (1967): Introduction
• Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (1968): $\S 1.2.2$
• T.S. Blyth: Set Theory and Abstract Algebra (1975): $\S 3$
• W.A. Sutherland: Introduction to Metric and Topological Spaces (1975): Notation and Terminology
• K.G. Binmore: Mathematical Analysis: A Straightforward Approach (1977)... (previous)... (next): $\S 2.9$
• Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): $\S 6$
• H. Jerome Keisler and Joel Robbin: Mathematical Logic and Computability (1996): Appendix $\text{A}.9$