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This entry defines Grothendieck topologies using sieves.
However, in the original definition (Michael Artin’s seminar notes “Grothendieck topologies”), a Grothendieck topology on a category $C$ is defined as a set $T$ of coverings.
More precisely (to cite from Artin’s notes), a Grothendieck topology is defined as families of maps $\{\phi_i\colon U_i\to U\}_{i\in I}$ such that
for any isomorphism $\phi$ we have $\{\phi\}\in T$;
if $\{U_i\to U\}\in T$ and $\{V_{i,j}\to U_i\}\in T$ for each $i$, then $\{V_{i,j}\to U\}\in T$;
if $\{U_i\to U\}\in T$ and $V\to U$ is a morphism, then $U_i\times_U V$ exist and $\{U_i\times_U V\to V\}\in T$.
This is almost identical to the current definition of Grothendieck pretopology, except that in Artin’s definition only the relevant pullbacks are required to exist.
It seems to me that the original definition by Artin is the one used most often in algebraic geometry.
Moved the content from historical note on Grothendieck topology here.
Added names for the axioms. Added a reference to SGA 4.
I removed the following axiom from the definition of a Grothendieck topology, since it is redundant (i.e., implied by the other axioms).
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