Rightopen

Rightopen refers to a type of interval on the real number line that is closed on the left endpoint and open on the right endpoint. The standard finite example is [a,b) where a < b; the interval contains every x with a ≤ x < b but does not contain b. In this sense it is left-closed, right-open. The term is sometimes used interchangeably with half-open intervals, though exact naming can vary by author.

Rightopen intervals are fundamental in several areas of analysis and topology. They frequently appear in measure theory, where the length of [a,b) is defined as b − a, and in the construction of Lebesgue measure. In topology, [a,b) serves as a basic building block for the lower limit topology (also called the Sorgenfrey line), whose basis consists of all half-open intervals [a,b). This distinguishes it from the standard topology on the real line, which uses open intervals (a,b) as a basis.

Properties commonly noted for [a,b) include that it is connected (as an interval in the real numbers), not compact in the usual topology of R since it is not closed, and it is a closed set only when the interval is unbounded on the left or right in extended contexts. If a or b is infinite, variants such as [a,∞) arise, which are left-closed and right-unbounded in the extended real line sense.

Rightopen intervals also play a role in analysis of functions defined on half-open domains, and they are often used in defining partitions of intervals for numerical methods and integration schemes that require consistent endpoint handling.