On $Q$ sets

William G. Fleissner, Arnold W. Miller
1980 Proceedings of the American Mathematical Society  
A Q set is an uncountable set X of the real line such that every subset of X is an F" relative to X. It is known that die existence of a Q set is independent of and consistent with the usual axioms of set theory. We show that one cannot prove, using the usual axioms of set theory: 1. If X is a Q set men any set of reals of cardinality less than the cardinality of X is a Q set. 2. The union of a Q set and a countable set is a Q set. The existence of a Q set is a fundamental question of set
more » ... considered by Hausdorff [1], Sierpinski [2] and Rothberger [3] over thirty years ago, and by many others since [4]-[7], [11] . A Q set is an uncountable subset X of the real line R such that every subset of X is an Fa relative to X. Precisely, for every A a X, there are countably many closed subsets Hn, n £ w, of R such that U {//": n £ w} n X = A.
doi:10.1090/s0002-9939-1980-0550513-4 fatcat:7jgqwzsfmzgupfgcwiaqhqkk4a