Paracompactness and Product Spaces

A topological space is called paracompact (see [2 J) if (i) it is a Hausdorff space (satisfying the T2 axiom of [l]), and (ii) every open covering of it can be refined by one which is "locally finite" ( = neighbourhood-finite; that is, every point of the space has a neighbourhood meeting only a finite number of sets of the refining covering). J. Dieudonné has proved [2, Theorem 4] that every separable metric ( = metrisable) space is paracompact, and has conjectured that this remains true without separability. We shall show that this is indeed the case. In fact, more is true; paracompactness is identical with the property of "full normality" introduced by J. W. Tukey [5, p. 53]. After proving this (Theorems 1 and 2 below) we apply Theorem 1 to obtain a necessary and sufficient condition for the topological product of uncountably many metric spaces to be normal (Theorem 4). For any open covering W~ {Wa) of a topological space, the star (x, W) of a point x is defined to be the union of all the sets Wa which contain x. The space is fully normal if every open covering V of it has a A-refinement" W—that is, an open covering for which the stars (xy W) form a covering which refines V.