Two remarks on A. Gleason's factorization theorem
Bulletin of the American Mathematical Society
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
... 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. The example. Let Z be a compact Hausdorff space which is first countable but not metrizable. Such a space exists by [l, §2, Exercise 13]. Since Z is completely regular, Z is homeomorphic to a compact subset U of a product X of copies of [0,1]. Let/: U-+Ube the identity. Assume that f -g o w\ u, with ir a countable projection and g: TT(U)->U continuous, and argue for a contradiction. Since countable products of separable metric spaces are separable metric, ir(U) is separable metric. Hence U is a continuous image of a separable metric space. But a cosmic metric space is metrizable whenever it is compact by [3, p. 994, (C) for cosmic spaces]. This contradicts the assumptions on Z. 2. A factorization theorem. The above example shows that the following result does not hold longer when Y is not metrizable. THEOREM. If Z is any subset of a product of arbitrary uniform spaces A MS Subject Classifications. Primary 5425, 5460; Secondary 5440.