It is known that all band preserving operators acting in a universally complete K-space are regular if and only if the K-space is locally one-dimensional. In addition, a K-space with base B is locally one-dimensional if and only if R^=R in V(B). It seems to have been unknown so far whether there exist nondiscrete locally one-dimensional K-spaces. In the present note we give a positive answer to the question. As an auxiliary result, we establish that a K-space is locally one-dimensional if and only if its base is σ-distributive.

Keywords:locally one-dimensional K-space, discrete K-space, σ-distributive Boolean algebra, σ-inductive Boolean algebra, atomic Boolean algebra, regular operator, real numbers in a Boolean-valued universe.