A sheaf on a topological space 
is a presheaf 
whose local sections can
be uniquely glued. More precisely, for every open
covering 
of an open set 
, it satisfies the following conditions:
1. If two sections 
have equal images under all restriction
maps to 
,
then 
.
2. If sections 
agree on every intersection 
, then there is a
section 
whose image under the restriction
map to each 
is 
.
For example, let 
be an algebraic variety over an algebraically
closed field 
. If 
denotes the ring of regular
functions from 
to 
, then with the usual restriction
maps, 
is a sheaf called the sheaf of regular
functions on 
.
The sheaf of continuous real-valued functions on any topological space and the sheaf of smooth real-valued functions on a smooth manifold are defined similarly.
