interval

Loosely speaking, an interval is a part of the real numbers that start at one number and stops at another number. For instance, all numbers greater that $1$ and smaller than $2$ form in interval. Another interval is formed by numbers greater or equal to $1$ and smaller than $2$. Thus, when talking about intervals, it is necessary to specify whether the endpoints are part of the interval or not. There are then four types of intervals with three different names: open, closed and half-open. Let us next define these precisely.

1. The open interval contains neither of the endpoints. If $a$ and $b$ are real numbers, then the open interval of numbers between $a$ and $b$ is written as $(a,b)$ and

2. The closed interval contains both endpoints. If $a$ and $b$ are real numbers, then the closed interval is written as $[a,b]$ and

   $$[a,b]=\{x\in ℝ\mid a\le x\le b\}.$$

3. A half-open interval contains only one of the endpoints. If $a$ and $b$ are real numbers, the half-open intervals $(a,b]$ and $[a,b)$ are defined as

   $(a,b]$ $=$

   $\mathrm{\hspace{0.17em}}[a,b)$ $=$

Note that this definition includes the empty set as an interval by, for example, taking the interval $(a,a)$ for any $a$.

An interval is a subset $S$ of a totally ordered set $T$ with the property that whenever $x$ and $y$ are in $S$ and then $z$ is in $S$. Applied to the real numbers, this encompasses open, closed, half-open, half-infinite, infinite, empty, and one-point intervals. All the various different types of interval in $ℝ$ have this in common. Intervals in $ℝ$ are connected under the usual topology.

There is a standard way of graphically representing intervals on the real line using filled and empty circles. This is illustrated in the below figures:

The logic is here that a empty circle represent a point not belonging to the interval, while a filled circle represents a point belonging to the interval. For example, the first interval is an open interval.

Infinite intervals

If we allow either (or both) of $a$ and $b$ to be infinite, then we define

$(a,\mathrm{\infty })$ $=$ $\mathrm{\{}x\in ℝ\mid x>a\},$

$\mathrm{\hspace{0.17em}}[a,\mathrm{\infty })$ $=$ $\mathrm{\{}x\in ℝ\mid x\ge a\},$

$(-\mathrm{\infty },a)$ $=$

$(-\mathrm{\infty },a]$ $=$ $\mathrm{\{}x\in ℝ\mid x\le a\},$

$(-\mathrm{\infty },\mathrm{\infty })$ $=$ $ℝ.$

The graphical representation of infinite intervals is as follows:

Note on naming and notation

In [1, 2], an open interval is always called a segment, and a closed interval is called simply an interval. However, the above naming with open, closed, and half-open interval seems to be more widely adopted. See e.g. [3, 4, 5]. To distinguish between $[a,b)$ and $(a,b]$, the former is sometimes called a right half-open interval and the latter a left half-open interval [6]. The notation $(a,b)$, $[a,b)$, $(a,b]$, $[a,b]$ seems to be standard. However, some authors (especially from the French school) use notation $]a,b[$, $[a,b[$, $]a,b]$, $[a,b]$ instead of the above (in the same ). Bourbaki, for example, uses this notation.

This entry contains content adapted from the Wikipedia article http://en.wikipedia.org/wiki/Interval_(mathematics)Interval (mathematics) as of November 10, 2006.

The metapost code for the figures can be found http://aux.planetmath.org/files/objects/4446/here.

Title	interval
Canonical name	Interval
Date of creation	2013-03-22 13:44:58
Last modified on	2013-03-22 13:44:58
Owner	PrimeFan (13766)
Last modified by	PrimeFan (13766)
Numerical id	16
Author	PrimeFan (13766)
Entry type	Definition
Classification	msc 12D99
Classification	msc 26-00
Classification	msc 54C30
Related topic	OpenSetsInMathbbRnContainsAnOpenRectangle
Related topic	LineSegment
Related topic	CircularSegment
Defines	open interval
Defines	closed interval
Defines	half-open interval
Defines	right half-open interval
Defines	left-half-open interval
Defines	segment