Two remarks on A. Gleason's factorization theorem

The theorem of A. Gleason [2, vii.23] asserts that every continuous map ƒ from an open subset U of a product X of separable topological spaces into a Hausdorff space Y whose points are Gs-sets has the form goir\uy where ir is a countable projection of X and g: ir(U)—> Y is continuous. A natural question is to find what other "pleasant" subsets U of X have the above factorization property. The most plausible ones are compact subsets: for, if UQX is compact and f = goTr\u with ƒ continuous, then g must be continuous since ir\ u is a closed map (being continuous on a compact space). The first part of this note rejects this conjecture by giving an example of a compact subset of a product of copies of the unit interval, without the factorization property. In the second part, it is proved that the factorization f = g o ir\u always holds whenever ƒ is uniformly continuous and the range metric. This result implies an open mapping theorem for continuous linear mappings on products of Fréchet spaces.