# Intuition for pseudo-points and the inductive step in Johnstone's proof of Deligne's completeness theorem

In Johnstone's Topos Theory appears the following lemma.

7.41 Lemma. Let $$P$$ be a pseudo-point of $$\mathsf C$$, $$X$$ a $$J$$-sheaf on $$\mathsf C$$, and $$x,y$$ two distinct element of $$P(X)$$. Let $$(V_j\to V)$$ be a finite $$J$$-covering family in $$\mathsf C$$, and $$v$$ and element of $$P(h_V)$$. There there exists a refinement $$Q$$ of $$P$$ such that (a) the images of $$x,y$$ under the natural map $$P(X)\to Q(X)$$ are distinct, and (b) the image of $$v$$ in $$Q(h_V)$$ is in $$\bigcup _j\; \operatorname{im}(Q(h_{V_j})\to Q(h_V))$$.

So the refinement continues to "separate" $$x,y$$, which have to do with the sheaf $$X$$ (but not the covering family $$(V_j\to V)$$), but has the additional property (b), which seems to be only about representables associated to the covering family $$(V_j\to V)$$ (and not the sheaf $$X$$).

Questions.

1. What is a nice conceptual intuition for the notion of a pseudo-point? Is there a geometric intuition?
2. What's the idea behind property (b)?

This lemma seems to be topological in nature, but I just can't seem to unpack the content of property (b) of the refinement.

The idea behind the notion of pseudo-point is very simple; it is just a description of a point using only the site. Since a point of a topos is by Diaconescu's theorem the same as a continuous flat functor from the site to $$\mathcal{S}et$$, when the site has finite limits this is the same as a filtered colimit of representable functors that in addition sends covering families to jointly surjective arrows in $$\mathcal{S}et$$. Therefore, one can identify the opint with a filtered diagram $$(U_i)_{i \in I}$$ in the site, which is precisely what Johnstone calls a pseudo-point.
As for property (b), it is just the inductive condition that ensures that in the end covering families are sent to jointly surjective arrows in $$\mathcal{S}et$$. Indeed, the continuous flat functor corresponding to the pseudo-point, evaluated in $$C$$, is precisely the filtered colimit $$\lim [U_i, C]$$, and condition (b) translates to a refinement of the filtered diagram to ensure the property above.