Halfopen

Halfopen, or half-open, is a term used in mathematics and computer science to describe a bound interval that includes one endpoint and excludes the other. It is most commonly encountered with intervals on the real number line and with ranges in programming.

In mathematics, a half-open interval is denoted [a, b) or (a, b], depending on which endpoint is included. It contains all x with a ≤ x < b (or a < x ≤ b). It is distinct from open intervals (a, b) and closed intervals [a, b]. In the real numbers with the usual topology, half-open intervals are bounded but not closed, and hence not compact. They are used in partitioning a domain, in defining integration limits, and in various constructions within measure theory and analysis.

In computing, many languages adopt half-open ranges, with an inclusive lower bound and an exclusive upper bound. Examples include Python’s range(n), which yields 0 through n−1, and slicing operations on sequences, which use a start inclusive and end exclusive convention. This approach simplifies arithmetic for lengths and iteration, and helps avoid off-by-one errors when concatenating adjacent ranges or segments.

Other terminology sometimes overlaps with the concept; half-open is often described as half-closed or semi-open in certain mathematical or computational contexts. The core idea remains a boundary that is included on one side and excluded on the other.