= Open cover criterion for a local sheaf topos
{title2=$\operatorname{Sh}(X)\text{ local}\Longleftrightarrow\exists x\text{ whose only open neighborhood is }X$}
Equivalently every open cover of $X$ contains $X$ as a member, so the top element of its open-set lattice is supercompact in the frame sense. Necessity follows from preservation of <coproducts> and <epimorphisms> by global sections. Conversely the union of all proper opens misses a point $x$, and the <stalk functor> there equals global sections. Its <right adjoint> is a <skyscraper sheaf of sets>. Empty spaces fail; a T1 space satisfies the condition exactly when it is a singleton.
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