A topological basis for a topology 
on a set 
is a collection 
such that every open set
in 
is a union of elements of 
. For the usual topology on the
real numbers, the collection 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).
