Basis for a Topology
A basis (or base) for a topology on a set is a collection of open sets (the basis elements) such that every open set in is the union or finite intersection of members of .[more]
Equivalently, a collection of open sets is a basis for a topology on if and only if it has the following properties:
1. For each , there is at least one basis element containing .
2. If and , then there is a basis element containing such that .
The set of all open disks contained in an open square form a basis. Drag the point within the square; then drag the centers of the disks and change their radii as needed to illustrate property 2 of a basis.[less]
 E. J. Borowski and J. M. Borwein, Collins Dictionary of Mathematics, HarperCollins Publishers, 1989 p. 46.