A sequentially closed countable dense subset of 𝐼^{𝐼}

Abstract

These sets have cardinality 2C and c, respectively. In [5], we raised the question of the existence of a countable such set. We here answer this question affirmatively by showing that P contains a countable dense subset that contains no nontrivial convergent sequences at all. Separability of P is well known, and follows, for instance, from a more general result of Pondiczery [4] (see also Marczewski [3]). Let JER be a maximal subset of irrationals linearly independent over the integers. Since / and / have the same cardinality, P and IJ are homeomorphic, and so our goal will be reached by exhibiting a countable dense subset of IJ having no nontrivial convergent sequences in it. Let 5 be the set {xn}neN in IJ such that