Editing Grothendieck topology (section)

==== Axioms ====
The conditions we impose on a Grothendieck topology are:

* (T 1) (Base change) If ''S'' is a covering sieve on ''X'', and ''f'': ''Y'' → ''X'' is a morphism, then the pullback ''f''<sup><math>\ast</math></sup>''S'' is a covering sieve on ''Y''.
* (T 2) (Local character)  Let ''S'' be a covering sieve on ''X'', and let ''T'' be any sieve on ''X''.  Suppose that for each object ''Y'' of ''C'' and each arrow ''f'': ''Y'' → ''X'' in ''S''(''X''), the pullback sieve ''f''<sup><math>\ast</math></sup>''T'' is a covering sieve on ''Y''.  Then ''T'' is a covering sieve on ''X''.
* (T 3) (Identity) Hom(&minus;, ''X'') is a covering sieve on ''X'' for any object ''X'' in ''C''.

The base change axiom corresponds to the idea that if {''U<sub>i</sub>''} covers ''U'', then {''U<sub>i</sub>'' ∩ ''V''} should cover ''U'' ∩ ''V''.  The local character axiom corresponds to the idea that if {''U<sub>i</sub>''} covers ''U'' and {''V<sub>ij</sub>''}<sub>''j <math>\in</math>J<sub>i</sub>''</sub> covers ''U<sub>i</sub>'' for each ''i'', then the collection {''V<sub>ij</sub>''} for all ''i'' and ''j'' should cover ''U''.  Lastly, the identity axiom corresponds to the idea that any set is covered by itself via the identity map.