A topological basis is a subset 
of a set 
in which all other open sets can
be written as unions or finite intersections
of 
.
For the real numbers, the set
of all open intervals is a basis.
Stated another way, if 
is a set, a basis for a topology
on 
is a collection 
of subsets of 
(called basis elements) satisfying the following properties.
1. For each 
,
there is at least one basis element 
containing 
.
2. If 
belongs to the intersection of two basis elements 
and 
, then there is a basis element 
containing 
such that 
.
(Munkres 2000).
