On product of shape and a question of Sher

In this paper the products in various shape categories are investigated. In the weak shape category defined by Borsuk, for arbitrary metrizable spaces X and Y there exists always the product Sh^X) X Sh^). In the shape category in the sense of Fox, if X is a pointed FANR and Y is an arbitrary metrizable space, there exists the product Sh F (X) X SMF) and the relation Sh F (X) X Sh F (Y) = Sh^X X Y) holds. In the proper shape category in the sense of Ball and Sher, the product does not exist generally. If X is a compactum and Y is a locally compact metrizable space, the proper shape of the product space X X Y is determined uniquely by the proper shapes of X and Y.