TWO REMARKS ON A. GLEASON'S FACTORIZATION THEOREM
BY GIOVANNI VIDOSSICH
Communicated by Leonard Gillman, August 8, 1969
The theorem of A. Gleason [2, vii.23] asserts t h a t 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" sub-sets 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).
T h e first part of this note rejects this conjecture by giving an example of a compact subset of a product of copies of the unit inter-val, without the factorization property. In the second part, it is proved t h a t 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.
1. 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 ] . L e t / : U-+Ube the iden-tity. Assume t h a t f — g o w\ u, with ir a countable projection and g: TT(U)—>U continuous, and argue for a contradiction. Since count-able products of separcount-able metric spaces are separcount-able 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. T h e above example shows t h a t the following result does not hold longer when Y is not metrizable.
T H E O R E M . If Z is any subset of a product of arbitrary uniform spaces
A MS Subject Classifications. Primary 5425, 5460; Secondary 5440.
Key Words and Phrases. Products of uniform spaces, Banach spaces, countable projections, factorizations upon countable projections, open mapping theorem, prod-ucts of Banach spaces.
A. GLEASON FACTORIZATION THEOREM 371 Xa ( a £ i ) into a metric space F, then every uniformly continuous ƒ: Z—+Y has the form goir\z with ir a countable projection and g uni-formly continuous.
P R O O F . By the uniform continuity of/, for each integer n^l there are a finite subset AnQA and uniform covers "Ma of Xa (aE:An) such t h a t
d(f(x),f(y)) £ 1/n
whenever x, yÇzZ have the coordinates corresponding to a(~An near of order 'll*. P u t C = U"«i An and ir the countable projection (#a)«eA
—>(xa)aeC' For every xÇzir{Z), let zx be a point of Zr\ir~l(pc).Define
g: 7r(Z)—>Y by x-~*f(zx). If s', s " £ Z have the same image by 7r, then <*(ƒ(*')» f(z"))Sl/n for all w ^ l (since C3^4W), which implies
d(fW)ff(z"))=0, i.e. ƒ(*') =ƒ(*"). This means t h a t g is well defined. From the definition it follows f~g OT\Z- T h e equality ƒ = g o w \ z means t h a t two points of Z have the same image by ƒ whenever they have the same coordinates for a £ C . By this and CQ.An ( w è 1), g is uniformly continuous. Q.E.D.
COROLLARY. Let Xa ( a £ 4 ) , Y be arbitrary complete metrizable
topological vector spaces. Then every continuous linear map ƒ from I I «GA Xa onto Y is open.
P R O O F . Since a continuous linear map is uniformly continuous in
the standard uniformities of topological vector spaces, the above theorem implies t h a t f = gowy with TT a countable projection and g uniformly continuous. Since TT and ƒ are linear, g is linear. Since a countable product of complete metric spaces is complete metric, g is open by Banach homomorphism theorem. Since TT is open, ƒ must be also. Q.E.D.
R E F E R E N C E S
1. N. Bourbaki, Topologie générale. Livre III: Chapitre 9: Utilisation des nombres réels en topologie générale. Actualités Sci. Indust., no. 1045, Hermann, Paris, 1958. MR 30 #3439.
2. J. R. Isbell, Uniform spaces, Math. Surveys, no. 12, Amer. Math. Soc, Provi-dence, R.I., 1964. MR 30 #561.
3. E. Michael, ^spaces, J. Math. Mech. 15 (1966), 983-1002. MR 34 #6723.
