Open Set

A Set is open if every point in the set has a Neighborhood lying in the set. An open set of Radius $r$ and center ${\bf x}_0$ is the set of all points ${\bf x}$ such that $\vert{\bf x}-{\bf x}_0\vert < r$, and is denoted $D_r({\bf x}_0)$. In 1-space, the open set is an Open Interval. In 2-space, the open set is a Disk. In 3-space, the open set is a Ball.

More generally, given a Topology (consisting of a Set $X$ and a collection of Subsets $T$), a Set is said to be open if it is in $T$. Therefore, while it is not possible for a set to be both finite and open in the Topology of the Real Line (a single point is a Closed Set), it is possible for a more general topological Set to be both finite and open.

The complement of an open set is a Closed Set. It is possible for a set to be neither open nor Closed, e.g., the interval $(0,1]$.

See also Ball, Closed Set, Empty Set, Open Interval