Editing Grothendieck topology (section)

=== Small site associated to a topological space ===

We repeat the example that we began with above.  Let ''X'' be a topological space.  We defined ''O''(''X'') to be the category whose objects are the open sets of ''X'' and whose morphisms are inclusions of open sets.  Note that for an open set ''U'' and a sieve ''S'' on ''U'', the set ''S''(''V'') contains either zero or one element for every open set ''V''. The covering sieves on an object ''U'' of ''O''(''X'') are those sieves ''S'' satisfying the following condition:
*If ''W'' is the union of all the sets ''V'' such that ''S''(''V'') is non-empty, then ''W'' = ''U''.
This notion of cover matches the usual notion in point-set topology.

This topology can also naturally be expressed as a pretopology.  We say that a family of inclusions {''V''<sub>''α''</sub> <math>\sube</math> ''U''} is a covering family if and only if the union  <math>\cup</math>''V''<sub>''α''</sub> equals ''U''.  This site is called the '''small site associated to a topological space''' ''X''.