Halboffenen

Halboffenen is a term used in topology and real analysis to describe sets that are not fully open but are in a sense between open and closed. In Scandinavian languages the word translates to “half-open” or “semi-open.” A common concrete example on the real line with the usual topology is the half-open interval [a, b), which contains its left endpoint but not its right.

In the real line, [a, b) is not an open set and not a closed set: it does not contain a neighborhood of every point, and it does not contain all its limit points. Nevertheless, it has properties associated with openness: its interior is (a, b), and its closure is [a, b]. A useful perspective is that [a, b) is semi-open (or half-open) in the sense that it lies between the interior and the closure. More formally, a subset S of a topological space is sometimes called semi-open if S is contained in the closure of its interior, i.e., S ⊆ cl(int(S)).

In topology more broadly, half-open sets provide a way to construct and study intermediate notions of openness. For example, the lower limit topology on the real numbers uses the half-open intervals [a, b) as a basis, yielding a topology finer than the standard one. This illustrates how half-open sets serve both as simple examples and as building blocks for alternative topological structures. See also open set, closed set, semi-open set, half-open interval.