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# 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$(a,b)=\{x\in\mathbb{R}\mid a<x<b\}.$ 2. The closed interval contains both endpoints. If $a$ and $b$ are real numbers, then the

*closed interval*is written as $[a,b]$ and$[a,b]=\{x\in\mathbb{R}\mid a\leq x\leq b\}.$ 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$\displaystyle(a,b]$ $\displaystyle=$ $\displaystyle\{x\in\mathbb{R}\mid a<x\leq b\},$ $\displaystyle\,\![a,b)$ $\displaystyle=$ $\displaystyle\{x\in\mathbb{R}\mid a\leq x<b\}.$

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 $x<z<y$ 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 $\mathbb{R}$ 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:

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

$\displaystyle(a,\infty)$ | $\displaystyle=$ | $\displaystyle\{x\in\mathbb{R}\mid x>a\},$ | ||

$\displaystyle\,\![a,\infty)$ | $\displaystyle=$ | $\displaystyle\{x\in\mathbb{R}\mid x\geq a\},$ | ||

$\displaystyle(-\infty,a)$ | $\displaystyle=$ | $\displaystyle\{x\in\mathbb{R}\mid x<a\},$ | ||

$\displaystyle(-\infty,a]$ | $\displaystyle=$ | $\displaystyle\{x\in\mathbb{R}\mid x\leq a\},$ | ||

$\displaystyle(-\infty,\infty)$ | $\displaystyle=$ | $\displaystyle\mathbb{R}.$ |

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 order). Bourbaki, for example, uses this notation.

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

# References

- 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.

## Mathematics Subject Classification

12D99*no label found*26-00

*no label found*54C30

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## Attached Articles

## Corrections

address the empty set by Wkbj79 ✓

Suggested generalization by dfeuer ✓

subinterval by Wkbj79 ✘

subinterval by Wkbj79 ✘

## Comments

## Interval entry

Apparently all the files attached to this entry were deleted. Can anyone explain? Just a bad link somewhere due to the server move?

Cam

## Re: Interval entry

I don't think the files are deleted.

The aux.planetmath server just seems to be down.

See

http://planetmath.org/?op=getmsg&id=9813

## losing all your files (was: Interval entry)

mathcam wrote:

> Apparently all the files attached to this

> entry were deleted. Can anyone explain?

> Just a bad link somewhere due to the server move?

This happened to me with the Cantor set article. I now make sure I have saved copies of any attached files before I edit an article. The files are completely gone - they are not on aux.planetmath.org. The snapshots don't contain attached files (Why not?!?), so I haven't been able to recover them.

## Snapshots don't contain attached files?

> mathcam wrote:

>

> > Apparently all the files attached to this

> > entry were deleted. Can anyone explain?

> > Just a bad link somewhere due to the server move?

>

> This happened to me with the Cantor set article. I now make

> sure I have saved copies of any attached files before I edit

> an article. The files are completely gone - they are not on

> aux.planetmath.org. The snapshots don't contain attached

> files (Why not?!?), so I haven't been able to recover them.

that's bogus

## Re: Interval entry

So what's the solution here? The article won't render unless I either

1) Get the pictures back.

2) Delete the code for the pictures in the article.

I'd obviously prefer the first option. Steve, do you still have the link to those files somewhere?

Cam

## Re: Interval entry

I re-uploaded the files and the metapost source.

Now it renders.