Editing Grothendieck topology (section)

=== Motivation ===
The classical definition of a sheaf begins with a topological space <math>X</math>.  A sheaf associates information to the open sets of <math>X</math>.  This information can be phrased abstractly by letting <math>O(X)</math> be the category whose objects are the open subsets <math>U</math> of <math>X</math> and whose morphisms are the inclusion maps <math>V\rightarrow U</math> of open sets <math>U</math> and <math>V</math> of <math>X</math>.  We will call such maps ''open immersions'', just as in the context of [[scheme (mathematics)|scheme]]s.  Then a presheaf on <math>X</math> is a [[contravariant functor]] from <math>O(X)</math> to the category of sets, and a sheaf is a presheaf that satisfies the [[gluing axiom]] (here including the separation axiom).  The gluing axiom is phrased in terms of [[Cover (topology)|pointwise covering]], i.e., <math>\{U_i\}</math> covers <math>U</math> if and only if <math> \bigcup_i U_i = U</math>. In this definition, <math>U_i</math> is an open subset of <math>X</math>. Grothendieck topologies replace each <math>U_i</math> with an entire family of open subsets; in this example, <math>U_i</math> is replaced by the family of all open immersions <math>V_{ij} \to U_i</math>. Such a collection is called a ''sieve''. Pointwise covering is replaced by the notion of a ''covering family''; in the above example, the set of all <math>\{V_{ij}  \to U_i\}_j</math> as <math>i</math> varies is a covering family of <math>U</math>. Sieves and covering families can be axiomatized, and once this is done open sets and pointwise covering can be replaced by other notions that describe other properties of the space <math>X</math>.