Grothendieck topology on a topological space #
Define the Grothendieck topology and the pretopology associated to a topological space, and show that the pretopology induces the topology.
The covering (pre)sieves on
X are those for which the union of domains contains
site, Grothendieck topology, space
- nLab, Grothendieck topology
- [S. MacLane, I. Moerdijk, Sheaves in Geometry and Logic][MM92]
Implementation notes #
We define the two separately, rather than defining the Grothendieck topology as that generated by the pretopology for the purpose of having nice definitional properties for the sieves.
The pretopology associated to a space is the largest pretopology that generates the Grothendieck topology associated to the space.
The pretopology associated to a space induces the Grothendieck topology associated to the space.