A HOMOGENEOUS SPACE THAT IS ONE-DIMENSIONAL BUT NOT COHESIVE

We present a homogeneous, one-dimensional, almost zero-dimensional space that is not cohesive. We also show that a complete ho-mogeneous space is cohesive if and only if it is not zero-dimensional. Every space under consideration is assumed to be separable metric. A space is called cohesive if every point of the space has a neighbourhood that fails to contain nonempty clopen subsets of the space. The cohesion concept plays a crucial role in characterizing Erdős space and complete Erdős space, see Dijkstra and van Mill [3, 4, 5]. Clearly, a cohesive space is at least one-dimensional at every point but the converse is not valid. However, the following useful result was proved in Dijkstra and van Mill [4, Proposition 6.3]: Proposition 1. A topological group is cohesive if and only if it is not zero-dimensional. In this note we show that this result does not generalize from topological groups to arbitrary homogeneous spaces. Our counterexample is almost zero-dimensional, that is, every point of the space has a neighbourhood basis that consists of C-sets, which are sets that can be written as intersections of clopen sets. Erdős space and complete Erdős space were introduced by Erdős [8] and are universal elements of the class of almost zero-dimensional spaces; see [4, Theo-rem 5.13]. Cohesion is a particularly useful property when combined with almost zero-dimensionality; see [4, §6]. Our counterexample is also strongly σ-complete