PlanetMath: interval

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interval

(Definition)

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 and smaller than form in interval. Another interval is formed by numbers greater or equal to and smaller than . 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.

The open interval contains neither of the endpoints. If and are real numbers, then the open interval of numbers between and is written as and
$\displaystyle (a,b)=\{ x\in \mathbb{R}\mid a<x<b\}.$
The closed interval contains both endpoints. If and are real numbers, then the closed interval is written as and
$\displaystyle [a,b]=\{ x\in \mathbb{R}\mid a\le x\le b\}.$
A half-open interval contains only one of the endpoints. If and are real numbers, the half-open intervals and are defined as

$\displaystyle (a,b]$ $\displaystyle =$ $\displaystyle \{ x\in \mathbb{R}\mid a<x\le b\},$

$\displaystyle \![a,b)$ $\displaystyle =$ $\displaystyle \{ x\in \mathbb{R}\mid a\le x< b\}.$

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

An interval is a subset of a totally ordered set with the property that whenever and are in and then is in . 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 $\mathbb{R}$ have this in common. Intervals in $\mathbb{R}$ 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:

$\includegraphics{intervals.1.eps}$

$\includegraphics{intervals.2.eps}$

$\includegraphics{intervals.3.eps}$

$\includegraphics{intervals.4.eps}$

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

and

to be infinite, then we define

$\displaystyle (a,\infty)$	$\displaystyle =$	$\displaystyle \{ x\in \mathbb{R}\mid x> a\},$
$\displaystyle \![a,\infty)$	$\displaystyle =$	$\displaystyle \{ x\in \mathbb{R}\mid x\ge a\},$
$\displaystyle (-\infty, a)$	$\displaystyle =$	$\displaystyle \{ x\in \mathbb{R}\mid x< a\},$
$\displaystyle (-\infty, a]$	$\displaystyle =$	$\displaystyle \{ x\in \mathbb{R}\mid x\le a\},$
$\displaystyle (-\infty, \infty)$	$\displaystyle =$	$\displaystyle \mathbb{R}.$

The graphical representation of infinite intervals is as follows:

$\includegraphics{intervals.5.eps}$

$\includegraphics{intervals.6.eps}$

$\includegraphics{intervals.7.eps}$

$\includegraphics{intervals.8.eps}$

$\includegraphics{intervals.9.eps}$

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

and

, the former is sometimes called a right half-open interval and the latter a left half-open interval [6]. The notation

seems to be standard. However, some authors (especially from the French school) use notation

instead of the above (in the same order). Bourbaki, for example, uses this notation.

This entry contains content adapted from the Wikipedia article Interval (mathematics) as of November 10, 2006.

Bibliography

1: W. Rudin, Principles of Mathematical Analysis, McGraw-Hill Inc., 1976.
2: W. Rudin, Real and complex analysis, 3rd ed., McGraw-Hill Inc., 1987.
3: R. Adams, Calculus, a complete course, Addison-Wesley Publishers Ltd., 3rd ed., 1995.
4: L. Råde, B. Westergren, Mathematics Handbook for Science and Engineering, Studentlitteratur, 1995.
5: R.A. Silverman, Introductory Complex Analysis, Dover Publications, 1972.
6: S. Igari, Real analysis - With an introduction to Wavelet Theory, American Mathematical Society, 1998.

The metapost code for the figures can be found here.

"interval" is owned by PrimeFan. [ full author list (3) | owner history (4) ]

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See Also: open set in $\mathbb{R}^n$ contains an open rectangle, line segment

Also defines:

open interval, closed interval, half-open interval, right half-open interval, left-half-open interval, segment

This object's parent.

Attachments:: open and closed intervals have the same cardinality (Result) by mps
subinterval (Definition) by Wkbj79

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Cross-references: Wikipedia, adapted, Bourbaki, representation, point, represent, logic, circles, line, usual topology, connected, infinite, property, totally ordered set, subset, empty set, contains, closed, open, types, endpoints, necessary, number, real numbers
There are 305 references to this entry.

This is version 13 of interval, born on 2003-07-14, modified 2008-02-25.
Object id is 4446, canonical name is Interval.
Accessed 33045 times total.

Classification:

AMS MSC:	12D99 (Field theory and polynomials :: Real and complex fields :: Miscellaneous)
	26-00 (Real functions :: General reference works )
	54C30 (General topology :: Maps and general types of spaces defined by maps :: Real-valued functions)

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