Complete Erdős space is unstable

Abstract

It is proved that the countably infinite power of complete Erdős space does not contain arbitrarily small closed subsets that are one-dimensional at every point. This observation also produces solutions to several problems that were posed by Aarts, Kawamura, Oversteegen and Tymchatyn. In addition, we show that the original (rational) Erdős space does contain arbitrarily small closed sets that are one-dimensional at every point.