# open

### open interval

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.)

### open cover

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.

### open set

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 \(U\).