Definition:Open


 * Topology:
 * A set $U$ in a topological space $\left({S, \tau}\right)$ is open iff $U \in \tau$.
 * A mapping $f: T_1 \to T_2$ from a topological space $T_1$ to another $T_2$ is open iff it maps open sets in $T_1$ to open sets in $T_2$.
 * An open cover is a cover consisting of open sets.
 * An open neighborhood is a neighborhood which is an open set.


 * Metric spaces:
 * A set $U$ in a metric space $\left({A, d}\right)$ is open iff every point in $U$ has an open $\epsilon$-ball lying entirely within $U$.
 * An open $\epsilon$-ball: the set of points of a metric space within $\epsilon \in \R_{>0}$ distance of a given point.


 * Complex Analysis:
 * A subset $U$ of the complex plane $\C$ is open iff every point in $U$ has a neighborhood lying entirely within $U$.


 * Real Analysis:
 * An open interval is a real interval which does not include its endpoints.