Productivity of Zariski-compactness for constructs of affine spaces

We study compactness for hereditary coreflective subconstructs X of SSET, the construct of affine spaces over the two point set S and with affine maps as morphisms, endowed with the Zariski closure operator z. We formulate necessary conditions for productivity of z-compactness. Moreover, if in X arbitrary products of quotients are quotients, then our conditions are also sufficient. We apply the results to some well-known subconstructs of SSET, in particular we investigate situations in which another sufficient condition for productivity of compactness, known as finite structure property for products (FSPP), is not fulfilled by the Zariski closure.