Sheaf
A sheaf is a presheaf with "something" added allowing us to define things locally. This task is forbidden for presheaves in general. Specifically, a presheaf 
on a topological space 
is a sheaf if it satisfies the following conditions:
1. if 
is an open set, if ![<span style=]()
{U_i}" src="https://mathworld.wolfram.com/images/equations/Sheaf/Inline4.gif" style="height:15px; width:24px" /> is an open covering of 
and if 
is an element such that 
for all 
, then 
.
2. if 
is an open set, if ![<span style=]()
{U_i}" src="https://mathworld.wolfram.com/images/equations/Sheaf/Inline11.gif" style="height:15px; width:24px" /> is an open covering of 
and if we have elements 
for each 
, with the property that for each, 
, 
, then there is an element 
such that 
for all 
.
The first condition implies that 
is unique.
For example, let 
be a variety over a field 
. If 
denotes the ring of regular functions from 
to 
then with the usual restrictions 
is a sheaf which is called the sheaf of regular functions on 
.
In the same way, one can define the sheaf of continuous real-valued functions on any topological space, and also for differentiable functions.
REFERENCES:
Godement, R. Topologie Algébrique et Théorie des Faisceaux. Paris: Hermann, 1958.
Hartshorne, R. Algebraic Geometry. New York: Springer-Verlag, 1977.
Iyanaga, S. and Kawada, Y. (Eds.). "Sheaves." §377 in Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, pp. 1171-1174, 1980.
