Interval
المؤلف:
Gemignani, M. C
المصدر:
Elementary Topology. New York: Dover, 1990.
الجزء والصفحة:
...
22-7-2021
1974
Interval
An interval is a connected portion of the real line. If the endpoints 
and 
are finite and are included, the interval is called closed and is denoted ![[a,b]](https://mathworld.wolfram.com/images/equations/Interval/Inline3.gif)
. If the endpoints are not included, the interval is called open and denoted 
. If one endpoint is included but not the other, the interval is denoted 
or ![(a,b]](https://mathworld.wolfram.com/images/equations/Interval/Inline6.gif)
and is called a half-closed (or half-open interval).
An interval ![[a,a]](https://mathworld.wolfram.com/images/equations/Interval/Inline7.gif)
is called a degenerate interval.
If one of the endpoints is 
, then the interval still contains all of its limit points, so 
and ![(-infty,b]](https://mathworld.wolfram.com/images/equations/Interval/Inline10.gif)
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[](https://mathworld.wolfram.com/images/equations/Interval/Inline11.gif)
for an open interval and 
or ![]a,b]](https://mathworld.wolfram.com/images/equations/Interval/Inline13.gif)
for a half-closed interval is sometimes also used.
A non-empty subset 
of 
is an interval iff, for all 
and 
, 
implies 
. If the empty set is considered to be an interval, then the following are equivalent:
1. 
is an interval.
2. 
is convex.
3. 
is star convex.
4. 
is pathwise-connected.
5. 
is connected.
REFERENCES:
Gemignani, M. C. Elementary Topology. New York: Dover, 1990.
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