open sets

light version:
a set U of real numbers is said to be open fif for each x in U, there exists an open interval (a,b) such that x is in (a,b) and (a,b) is a subset of U.
medium version:
a set U of a given set X, relative to a distinguished collection B of subsets of X, is said to be open fif for each x in U, there exists a member V of B such that x is in V and V is a subset of U.
heavy version:
a set U of a given set X, relative to a distinguished collection S of subsets of X, is said to be open fif for each x in U, there exists a member V of B such that x is in V and V is a subset of U, where B is generated from S by taking all possible arbitrary unions and all possible finite intersections.
Here is a web page concerned with this.
keywords: General Topology