Connectedness in the scale of uniform subspaces of R

Abstract

Kent [4] showed that each uniform space (S, u) could be embedded in a complete, uniform lattice, called the scale of (S, u). The scale was first introduced by Bushaw [3] for studying stability in topological dynamics. In [5], the notions of connectedness and local connectedness were studied. This note is a follow-up of [5]; the purpose being to characterise the uniform subspaces of the reals, R, which have connected (locally connected) scales. The reader is asked to refer to [5] for definitions and notation not given here.