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https://en.wikipedia.org/wiki/Grothendieck_topology#CITEREFArtin1962
Grothendieck topology
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The term "Grothendieck topology" has changed in meaning.
In Artin (1962) it meant what is now called a Grothendieck pretopology, and some authors still use this old meaning.
Giraud (1964) modified the definition to use sieves rather than covers.
Much of the time this does not make much difference, as each Grothendieck pretopology determines a unique Grothendieck topology, though quite different pretopologies can give the same topology.

Contents
1 Overview
2 Definition
2.1 Motivation
2.2 Sieves
2.3 Grothendieck topology
2.3.1 Axioms
2.3.2 Grothendieck pretopologies
3 Sites and sheaves
4 Examples of sites
4.1 The discrete and indiscrete topologies
4.2 The canonical topology
4.3 Small site associated to a topological space
4.4 Big site associated to a topological space
4.5 The big and small sites of a manifold
4.6 Topologies on the category of schemes
5 Continuous and cocontinuous functors
5.1 Continuous functors
5.2 Cocontinuous functors
5.3 Morphisms of sites