A note on universal sets for classes of countable G_δ'S

Abstract

In a recent article [2] D. G. Larman and C. A. Rogers proved the following two results in Descriptive Set Theory (where R = the space of real numbers): (1) There is no analytic set in the plane R^2, which is universal for the countable closed subsets of R; (2) there is no Borel set in R^2, which is universal for the countable G_δ subsets of R. Recall that, if b is a class of subsets of a space X, a set U ⊆ X × X is called universal for C if (ɑ) for each x є X, U_x = def {y : (x, y) U} є C, and (b) for each A є C there is an x such that A = U_x. (Larman and Rogers have also shown that in both cases coanalytic universal sets exist.)