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
ShW(X) ×ShW(Y ). 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 ShF(X)×ShF(Y ) and the relation ShF(X) ×ShF(Y ) =ShF(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 × Y is determined uniquely by the proper shapes of X and
Y .
