A subbasis for a topology 
on a set 
is a collection 
of open sets whose finite intersections form a topological
basis for 
. Here the empty intersection
is taken to be 
. Equivalently, 
consists of all unions of these
finite intersections
and is the coarsest topology containing 
(Sharifi, p. 15). The empty union
supplies the empty set.
If 
is a topological
space, a map 
is continuous iff 
is open
in 
for every 
.
This follows because inverse images preserve unions and finite intersections.
A subbasis for the Euclidean topology of the real line is formed by all intervals 
and 
: in fact a basis is formed by the open
intervals 
for 
.
For a fixed positive integer 
,
all 
-element subsets
of 
form a subbasis for the discrete
topology of the real line, since every singleton
set 
can be obtained as the intersection of the sets 
and 
.
A subbasis for the Zariski topology of the affine space 
is formed by the complement
sets of all irreducible affine varieties. This
follows applying de Morgan's laws when considering
that the open sets are the complement
sets of the affine varieties, and each of these
is the union of a finite number
of irreducible varieties.
