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

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

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 . 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.

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.

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