semilukket

Semilukket is a term used in mathematics, particularly in Norwegian, to describe a half-open or half-closed interval. Such a set includes one endpoint while excluding the other. The common notations are [a, b) and (a, b], where a and b are real numbers with a < b. These intervals are widely used in real analysis, measure theory, and related fields.

In the real numbers with the standard topology, a semilukket interval is neither open nor closed in general. For example, [0, 1) contains 0 but not 1, so it is not open, and it is not closed because its complement is not open. Despite this, semilukket intervals are simple to describe and manipulate algebraically; their length is b − a, and they are Lebesgue measurable. They are bounded when a < b and connected, since they form a single continuous block of points.

Variants and related terms: the concept is often called a half-open (or half-closed) interval in English. The two possibilities [a, b) and (a, b] are both considered semilukkede intervals in many Nordic sources. The notion extends beyond intervals to more general sets that are closed on one side and open on the other, though the standard example remains a half-open interval on the real line.

See also: open interval, closed interval, half-open interval, topology, Borel sets.