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A topological space is called paracompact (see [2 J) 1 if (i) it is a Hausdorff space (satisfying the T 2 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~ {W a ) of a topological space, the star (x, W) of a point x is defined to be the union of all the sets W a which contain x. The space is fully normal if every open covering V of it has a u A-refinement" W-that is, an open covering for which the stars (x y W) form a covering which refines V. THEOREM 1. Every fully normal Ti space is paracompact.Let S be such a space, and let V = { U a } be a given open covering of S. (We must construct a locally finite refinement of V. Note that S is normal [5, p. 49] and thus satisfies the T2 axiom.)There exists an open covering V 1 = { U 1 } which A-refines V, and by induction we obtain open coverings V n = { U n } of S such that

The following theorem is well known under the self-explanatory name of the "ham sandwich theorem".Given any three sets in space, each of finite outer Lebesgue measure (m*), there exists a plane which bisects all three sets, in the sense that the part of each set which lies on one side of the plane has the same outer measure as the part of the same set which lies on the other side of the plane.The usual proof is based on the following theorem of Borsuk.If is a continuous mapping of the n-sphere S" in Euclidean n-space R which is "antipodal" (i.e., diametrically opposite points of S map into points symmetric about the origin in R), then there is a point of S which maps into the origin of R .

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