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Interval


Interval

An interval is a connected portion of the real line. If the endpoints a and b are finite and are included, the interval is called closed and is denoted [a,b]. If the endpoints are not included, the interval is called open and denoted (a,b). If one endpoint is included but not the other, the interval is denoted [a,b) or (a,b] and is called a half-closed (or half-open interval).

An interval [a,a] is called a degenerate interval.

If one of the endpoints is +/-infty, then the interval still contains all of its limit points, so [a,infty) and (-infty,b] are also closed intervals. Intervals involving infinity are also called rays or half-lines. If the finite point is included, it is a closed half-line or closed ray. If the finite point is not included, it is an open half-line or open ray.

The non-standard notation ]a,b[ for an open interval and [a,b[ or ]a,b] for a half-closed interval is sometimes also used.

A non-empty subset X of R is an interval iff, for all a,b in X and c in R, a<=c<=b implies c in X. If the empty set is considered to be an interval, then the following are equivalent:

1. X is an interval.

2. X is convex.

3. X is star convex.

4. X is pathwise-connected.

5. X is connected.


See also

Bisection, Closed Interval, Half-Closed Interval, Interval Arithmetic, Limit Point, Line Segment, Open Interval, Pencil, Ray, Unit Interval Explore this topic in the MathWorld classroom

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References

Gemignani, M. C. Elementary Topology. New York: Dover, 1990.

Referenced on Wolfram|Alpha

Interval

Cite this as:

Weisstein, Eric W. "Interval." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Interval.html

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