We study the condition, on a connected and locally connected geometric morphism $p : \cal E \to \cal S$, that the canonical natural transformation $p_*\to p_!$ should be (pointwise) epimorphic - a condition which F.W. Lawvere 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.
Keywords: axiomatic cohesion, locally conected topos
2000 MSC: Primary 18B25, secondary 18A40
Theory and Applications of Categories, Vol. 25, 2011, No. 3, pp 51-63.