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 . 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  and is called a half-closed (or half-open interval).

An interval  is called a degenerate interval.

If one of the endpoints is , then the interval still contains all of its limit points, so  and  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  for an open interval and  or  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.