Could not include topos theory - contents
A connected site is a site satisfying sufficient conditions to make its topos of sheaves into a connected topos.
Let $C$ be a locally connected site; we say it is a connected locally connected site if it is also has a terminal object.
If $C$ is connected locally connected site, then the sheaf topos $Sh(C)$ is a locally connected topos and connected topos.
Being a locally connected site, we already know that we have a locally connected topos $(\Pi_0 \dashv \Delta \Delta \Gamma) : Sh(C) \to Set$. By the discussion there we need to check that $\Pi_0$ preserves the terminal object.
The terminal object in the site represents the terminal presheaf on $C$, which is the presheaf constant on the point. By the discussion at locally connected site we have that every constant presheaf is a sheaf over $C$, hence the terminal object of $Sh(C)$ is also represented by the terminal object in the site, and we just write “$*$” for all these terminal objects.
By the discussion there, the left adjoint $\Pi_0$ in the sheaf topos over a locally connected site is given by the colimit functor $\lim_\to : [C^{op}, Set]\to Set$. The colimit over a representable functor is always the point (this is the (co)-Yoneda lemma in slight disguise), hence indeed $\Pi_0 * = *$.
The category of open subsets $Op(X)$ of a locally connected and connected topological space $Op(X)$ with the standard open cover-coverage is a connected site. The terminal object is $X$ itself.
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