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.