Let $E$ be a (Grothendieck) topos, e.g. $E = \text{Sh}(X)$ for a topological space $X$. And let $p = (p^*, p_*):\text{Set}\to E$ be a point of $E$, is there a way to "puncture" $E$ in some sense? By "puncture" I mean a way to remove the point $p$ and still get a topos.

e.g. one may expect to get $\text{Sh}(X\backslash\{p\})$ if $E = \text{Sh}(X)$ for a topological space $X$ and $p \in X$ is a usual point. But it's not obvious to see how to do this at the level of topos.

I'm asking this question because I want to know if topoi are "unpuncturable" in some sense.

nota trivial thing. $\endgroup$ – h__ Oct 27 '16 at 4:30