Factoring continuous mappings defined on subspaces of topological products

Let Y be a subspace of the product space X = Pi(i is an element of I) X-i and f : Y -> Z be a continuous mapping. We present several conditions on Y and Z which guarantee the existence of the smallest (by inclusion) set J subset of I such that f depends on J. This happens, for example, if Y = X and Z is a T-0-space. We also show that such a set J subset of I exists if either Y is open and dense in X and Z is Hausdorff or Y contains the sigma-product sigma X(a) subset of X with center at some point a is an element of X and the space Z is regular. (C) 2020 Elsevier B.V. All rights reserved.