An interval of the real number line is said to be an open interval if it does not contain its endpoints. In interval notation this is denoted by using parentheses to enclose the numbers defining the interval. For example, the open interval defined by \( 0 < x < 1\) is denoted \((0,1)\). An interval containing only one of its endpoints is called half-open. (See: interval.)
A collection of open sets whose union contains a given set is called an open cover (or open covering) of the set.
open ball & open disk
The interior of a circle, excluding the circle itself (i.e., excluding the boundary) is called an open disk. In three or higher dimensional spaces we speak analogously of an open ball, which is the interior of a sphere excluding the surface of the sphere itself. Each is an example of a neighborhood of the point at the center of the disk or ball.
In topology, a subset \(U\) of a topological space \(X\) is said to be an open set if every element of \(U\) is an element of some open set of \(X\) that is contained in \(U\). In a metric space, A set \(U\) is open if for every element \(x\) of \(U\) we may find a positive distance \(\varepsilon\) such that the \(\varepsilon\)-neighborhood of \(x\) is a subset of \(U\).