The Generalized Borel Conjecture and Strongly Proper Orders

The Borel Conjecture is the statement that C = [R]<ü,i , where C is the class of strong measure zero sets; it is known to be independent of ZFC. The Generalized Borel Conjecture is the statement that C = [R]w-bounding orders. The central lemma is the observation that A. W. Miller's proof that the statement (*) "Every set of reals of power c can be mapped (uniformly) continuously onto [0,1] " holds in the iterated Sacks model actually holds in several other models as well. As a result, we show for example that (*) is not restricted by the presence of large universal measure zero (Un) sets (as it is by the presence of large C sets). We also investigate the er-ideal f = {X c R:X cannot be mapped uniformly continuously onto [0, 1]} and prove various consistency results concerning the relationships between ^, Un , and AFC (where AFC = {X CU:X is always first category}). These latter results partially answer two questions of J. Brown.