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The new set of families (for each object $X$) is called the sieve generated by the existing covers of $X$. One term for a Grothendieck pretopology is a basis for a Grothendieck topology, and different bases can give rise to the same Grothendieck topology. All of them, and the topology they generate, have the same sheaves.

See here for example.

Edit: Actually it is proposition C.2.1.9 in Johnstone's Sketches of an Elephant (Google books )

The new set of families (for each object $X$) is called the sieve generated by the existing covers of $X$. One term for a Grothendieck pretopology is a basis for a Grothendieck topology, and different bases can give rise to the same Grothendieck topology. All of them, and the topology they generate, have the same sheaves.
The new set of families (for each object $X$) is called the sieve generated by the existing covers of $X$. One term for a Grothendieck pretopology is a basis for a Grothendieck topology, and different bases can give rise to the same Grothendieck topology. All of them, and the topology they generate, have the same sheaves.