Negligible sets in Erdős spaces

Both spaces were introduced and shown to be one-dimensional but totally disconnected by Paul Erdős [9] in 1940. This result together with the obvious fact that E and Ec are homeomorphic to their squares make these spaces important examples in Dimension Theory. Both E and Ec are universal spaces for the class of almost zero-dimensional spaces; see [7, Theorem 4.15]. A subset of a space is called a C-set if it is an intersection of clopen sets. A space is called almost zerodimensional if every point has a neighbourhood basis consisting of C-sets. The spaces E, Ec, and also Ec were characterized by Dijkstra and van Mill [5,7,6] and Dijkstra [4]. Complete Erdős space plays a role in complex dynamics (Mayer [11], Aarts and Oversteegen [1]) and it can be represented by, for instance, end-point sets in R-trees (Kawamura, Oversteegen, and Tymchatyn [10]) or Polishable ideals (Dijkstra and van Mill [6]). The most important alternative representation of Erdős space is as the group of homeomorphisms of a topological manifold of dimension at least 2 that leave a countable dense set invariant (Dijkstra and van Mill [7, Theorem 10.2]). A subset A of a space X is called negligible if X \ A is homeomorphic to X . Kawamura, Oversteegen, and Tymchatyn [10] proved that σ -compacta and proper closed subsets are negligible in Ec. The authors proved in [7, Corollary 8.15] that proper closed subsets are negligible in E. In this paper we show that σ -compacta are also negligible in E. Dijkstra [4] proved that σ -compacta are negligible in Ec . It is not known whether proper closed subsets of E ω c are negligible. The main result in this paper improves on the negligibility of σ -compacta in Ec, as follows.