A set whose square can map onto a perfect set

Abstract

Assuming the continuum hypothesis, a set X of real numbers will be constructed such that every compact continuous image of X is countable, but XXX admits a uniformly continuous mapping onto the Cantor set. This completes (modulo the continuum hypothesis and the Ulam measure problem) the determination of whether a product or coproduct of m Boolean algebras or fields of sets can have an infinite free, or projective, or injective subor quotient object when the factors do not [1]. The concluding construction (modulo the hypothesis) is embarrassingly easy, but the result is unlike the others [1 ]; this is the only instance of a finite coproduct creating a remarkable subobject (viz. an infinite free subfield), and there is no instance among these of a finite product creating a remarkable quotient. The use made of the continuum hypothesis seems slight. However, Sierpinski has pointed out that the weaker result that there is a set of real numbers of the power of the continuum admitting no continuous mapping onto [0, 1] (C5 in [2]) has not been established without the hypothesis. On the continuum hypothesis, that is a by-product of an important construction of Lusin, the subject of Chapter II of [2]. We shall find X in the Cantor set C considered as an infinite power of the compact group Z2 (written additively). By a lemma of Lavrentiev (Theorem 99 of [3]), any continuous mapping of a subset of C onto [0, 1] can be extended over a Ga subset of C. There are only c (the power of the continuum) Ga sets, and each has only c continuous mappings to [0, 1]. Hence we can index all these mappingsfa by ordinals of smaller cardinal than c, and index similarly the points xa of C. For each fa, the inverse images of points are c disjoint subsets of C each closed in their union; so except for countably many, they are nowhere dense. By the continuum hypothesis, C is not a union of fewer than c such sets. There is no difficulty in building up sets S, T, no subset of either of which is mapped onto [0, 1] by any fa, but with every xa representable as sa+ta, Sa in S and ta in T. Carry along expanding sets Ha disjoint from S, Ka disjoint from T (Ho and Ko empty). Arrived