The category $\textbf{Set}$ can be given a Grothendieck topology where the covering families are jointly surjective families of set inclusions $\{X\_i\stackrel{\phi\_i}{\hookrightarrow} X\}\in\mathrm{Cov}(X)$, $X\in\mathrm{ob}(\textbf{Set})$.



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> Are there any other Grothendieck topologies on $\textbf{Set}$, not equivalent to the above one?
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Two Grothendieck topologies on $\mathcal{C}$ are *equivalent* when they give rise to sheaf categories which are equivalent as reflective subcategories of the category of presheaves on $\mathcal{C}$.



Are all Grothendieck topologies on Set equivalent?
(Much of this has basically been said by someone in the comments already.)


Here is a way of making examples of topologies on Set. Let $\mathcal C$ be a class of sets. Define a topology on Set by saying that a sieve $S$ on an object $X$ is a cover if and only if for every $Y\in\mathcal C$ every morphism $Y\to X$ belongs to $S$. An object is a point (in the sense that the only cover of that object is its maximal sieve) if it belongs to $\mathcal C$, and more generally if it is a retract of such an object. If we enlarge $\mathcal C$ to make it closed under retraction the topology is unchanged. Any presheaf that is a right Kan extension from (the full subcategory) $\mathcal C$ is a sheaf, and conversely any sheaf coincides with the Kan extension of its restriction. 


(All of that is valid for any category, not just for Set.)


In the case when $\mathcal C$ consists of just a singleton, this topology on Set is the canonical one, where the sheaves are the representable funtors. If $\mathcal C$ has at least one nonempty set in it then the sheaf is subcanonical. 


In general a class of sets closed under retraction must be either: all sets with cardinality less than a fixed cardinal, or all non-empty sets with cardinality less than a fixed cardinal. For example, $\mathcal C$ might be all nonempty sets having at most $n$ elements, or all nonempty finite sets, or all nonempty countable sets, or all nonempty sets (or any of these together with the empty set).


I suppose there are examples of topologies on Set not of this kind. That is, I suppose that there is a topology $T$ on Set such that $T$ does not coincide with the largest one that has the same points as $T$. Does somebody know?


EDIT: I realize now that of course there are perfectly everyday examples not of that kind. For example, the topology generated by finite covers in the ordinary sense. ($X$ is covered by ${Y\_i\to X}$ if $X$ is the union of $Y\_1,\dots ,Y\_n$.)


