Remarks on Punctual Local Connectedness

We study the condition, on a connected and locally connected geometric morphism p : E → S, that the canonical natural transformation p∗ → p! should be (pointwise) epimorphic — a condition which F.W. Lawvere [11] called the ‘Nullstellensatz’, but which we prefer to call ‘punctual local connectedness’. We show that this condition implies that p! preserves finite products, and that, for bounded morphisms between toposes with natural number objects, it is equivalent to being both local and hyperconnected.