Mapping a Set of Reals Onto the Reals
Arnold W. Miller
Journal of Symbolic Logic, Vol. 48, No. 3. (Sep., 1983), pp. 575-584.
AbstractIn this paper we show that it is consistent with ZFC that for any set of reals of cardinality the continuum, there is a continuous map from that set onto the closed unit interval. In fact, this holds in the iterated perfect set model. We also show that in this model every set of reals which is always of first category has cardinality less than or equal to $\omega_1$.