Editing Grothendieck topology (section)

=== Big site associated to a topological space ===

Let ''Spc'' be the category of all topological spaces.  Given any family of functions {''u''<sub>''α''</sub> : ''V''<sub>''α''</sub> → ''X''}, we say that it is a '''surjective family''' or that the morphisms ''u''<sub>''α''</sub> are '''jointly surjective''' if  <math>\cup</math> ''u''<sub>''α''</sub>(''V''<sub>''α''</sub>) equals ''X''.  We define a pretopology on ''Spc'' by taking the covering families to be surjective families all of whose members are open immersions.  Let ''S'' be a sieve on ''Spc''.  ''S'' is a covering sieve for this topology if and only if:
*For all ''Y'' and every morphism ''f'' : ''Y'' → ''X'' in ''S''(''Y''), there exists a ''V'' and a ''g'' : ''V'' → ''X'' such that ''g'' is an open immersion, ''g'' is in ''S''(''V''), and ''f'' factors through ''g''.
*If ''W'' is the union of all the sets ''f''(''Y''), where ''f'' : ''Y'' → ''X'' is in ''S''(''Y''), then ''W'' = ''X''.

Fix a topological space ''X''.  Consider the [[comma category]] ''Spc/X'' of topological spaces with a fixed continuous map to ''X''.  The topology on ''Spc'' induces a topology on ''Spc/X''.  The covering sieves and covering families are almost exactly the same; the only difference is that now all the maps involved commute with the fixed maps to ''X''.  This is the '''big site associated to a topological space''' ''X''.  Notice that ''Spc'' is the big site associated to the one point space.  This site was first considered by [[Jean Giraud (mathematician)|Jean Giraud]].