TY - JOUR
AU - Rogers, C. A.
AB - THE DESCRIPTIVE CHARACTER OF CERTAIN UNIVERSAL SETS By D. G. LARMAN and C. A. ROGERS [Received 30 May 1972] 1. Introduction If X and Y are spaces and ^ is a class of subsets of X, a set U in X x Y is said to be universal for the class ^ if lv) (a) for each y in Y the set U , of those points a;in l with xxy e U, belongs to ^ ; and (b) for each set C of ^, there is a point y of Y such tha t the corresponding lv) set U coincides with C. Using the axiom of choice, it is clear that a necessary and sufficient condition, for the existence of a set U universal for the class # , is that ^ be non-empty and have cardinal not exceeding the cardinal of Y. The existence problem becomes interesting, when ^ is a class of 'topologically respectable' sets, and we ask that the universal set be itself 'topologically respectable'. For example, when X and Y coincide with the real line R, there are sets U in R universal for the open, closed, ^- , & -, analytic and co 
TI - The Descriptive Character of Certain Universal Sets
JO - Proceedings of the London Mathematical Society
DO - 10.1112/plms/s3-27.3.385
DA - 1973-10-01
UR - https://www.deepdyve.com/lp/wiley/the-descriptive-character-of-certain-universal-sets-s9ZBFcaSPh
SP - 385
EP - 401
VL - s3-27
IS - 3
DP - DeepDyve
ER -