TITLE: Is there a way to "puncture" a topos?
QUESTION [9 upvotes]: 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.

REPLY [5 votes]: A more general question is how do you form the image of a geometric morphism f : E -> F, as a subtopos of F, and how do you form its complement? Recall that if U is a subobject of 1 in F then it defines an open subtopos (given by the operator U => ? on subobjects of 1) and its closed complement F-U is given by the operator of union with U on subobjects of 1. The closure of the image of f has to be the complement of the suprema of all the subobjects U of 1 in F for which f*(U) is 0.
I have memories of a nice paper by Anders Kock about glueing which may yield more.