halfclosed

Halfclosed is an informal term used to describe a set or interval that is closed at one endpoint and open at the other. It is most commonly encountered in real analysis and topology when discussing intervals on the real line. The standard notations are [a,b) and (a,b], which are respectively closed at the left and open at the right, and open at the left and closed at the right. The inclusion or exclusion of endpoints is indicated by the brackets: [ includes the endpoint, ( excludes the endpoint.

In mathematics, half-closed intervals [a,b) or (a,b] are neither open nor closed in the real line, though they have well-defined closures and interiors. The left-closed, right-open interval [a,b) has boundary points a and b; it is bounded but not closed, and therefore not compact in R. The collection of all half-open intervals with various endpoints is frequently used as a basis for the standard topology on the real line, illustrating how open sets can be built from semi-open building blocks.

In practical terms, half-open intervals are widely used in computation and data structures, where half-open ranges help avoid overlaps when concatenating consecutive intervals. They are also common in time interval notation, where [start, end) includes the start moment but excludes the end moment.

Variants and terminology include half-open, half-closed, and semi-open; different communities may prefer different labels, but the underlying concept remains the same. See also interval notation, open interval, closed interval, and half-open interval.